Publication Date: 2025
Journal of Algebra and its Applications (17936829)
We change Chermak-Delgado measure slightly so that it can be used for compact groups as well. Corresponding to this new measure, a lattice of open subgroups is obtained. Then we prove that if this lattice is non-empty in a compact group G, then G has a characteristic open abelian subgroup N such that [G: N] ≤ [G: A]2 for every abelian subgroup A G. This is a generalization of the well-known Chermak-Delgado Theorem. © 2026 World Scientific Publishing Company.
Publication Date: 2025
IEEE Access (21693536)13pp. 159412-159421
We present a constructive lower bound for permutation codes under Kendall’s τ-metric that improves upon the classical Gilbert-Varshamov estimate for many parameters. For any length p > 5, we construct codes with minimum Kendall’s τ-distance 6 and size at least p!/p3. This result is achieved by proving that Reed-Solomon codes of length p and dimension p-3 have a minimum Lee distance of 6, and then applying a metric embedding from the Lee metric to the Kendall’s τ-metric. © 2013 IEEE.
Publication Date: 2025
IEEE Transactions on Information Theory (00189448)71(6)pp. 4156-4166
In order to overcome the challenges caused by flash memories and also to protect against errors related to reading information stored in DNA molecules in the shotgun sequencing method, the rank modulation method has been proposed. In the rank modulation framework, codewords are permutations. In this paper, we study the largest size P(n, d) of permutation codes of length n, i.e., subsets of the set Sn of all permutations on {1, . . . , n} with the minimum distance at least d ∈ {1, . . . ,(Formula presented)} under the Kendall τ-metric. By presenting an algorithm and two theorems, we improve the known lower and upper bounds for P(n,d) . In particular, we show that P(n,d)=4 for all n ≥ 6 and (Formula presented) (Formula presented) < d ≤ (Formula presented) (Formula presented) . Additionally, we prove that for any prime number n and integer r ≤ − (Formula presented) , P(n,3) ≤ (n-1)! (Formula presented) (Formula presented) . This result greatly improves the upper bound of P(n,3) for all primes n ≥ 37 . © 1963-2012 IEEE.
Publication Date: 2025
IEEE Access (21693536)13pp. 15339-15345
Let q=pr be a prime power, Fq be the finite field of order q and f(x) be a monic polynomial in Fq [x]. Set A:=Fq [x]/ < f(x) >. In this paper we continue the study (started by T. P. Berger and N. El Amrani) of A -codes of length l over A , i.e. A -submodules of Al. We introduce two types of unique generating sets, called type I and type II basis of divisors, for an A -code. Using this, we present a building-up construction so that one can obtain all distinct A -codes of length l, with their basis of divisors. We complete the classification for the special case l=2 and enumerate all the A -codes of length 2. As an example, we list all binary index-2 quasi-cyclic codes of lengths 16 and 32, and all ternary index-2 quasi-cyclic codes of lengths 6 and 18, which are best-known codes. © 2025 The Authors.
Publication Date: 2025
Ars Mathematica Contemporanea (18553974)pp. 1-21
In this paper, we introduce the concept of k-integral graphs. A graph Γ is called kintegral if the extension degree of the splitting field of the characteristic polynomial of Γ over rational field Q is equal to k. We prove that the set of all finite connected graphs with given algebraic degree and maximum degree is finite. 1-integral graphs are just integral ones, graphs all of whose eigenvalues are integer. We study 2-integral Cayley graphs over finite groups G with respect to Cayley sets which are a union of conjugacy classes of G. Among other general results, we completely characterize all finite abelian groups having a connected 2-integral Cayley graph with valency 2, 3, 4 and 5. Furthermore, we classify finite groups G for which all Cayley graphs over G with bounded valency are 2-integral. cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
Publication Date: 2025
Cryptography and Communications (19362447)17(4)pp. 1075-1091
In this paper, we study the largest size A(n, d) of permutation codes of length n, i.e., subsets of the set Sn of all permutations on n letters with the minimum distance at least d under the Hamming metric. In Abdollahi et al. (Cryptogr. Commun. 15, 891–903 2023) we have developed a method using the representation theory of symmetric groups to find upper bounds on the size of permutation codes in Sn with the minimum distance of d under the Kendall τ-metric. The latter method is used for the permutation codes under the metric induced by Cayley graphs of Sn. Since the metric induced by any Cayley graph of Sn is not equivalent to the Hamming metric, we can not use the method for the Hamming metric. In this paper we find a trick by which we can again use the method to find upper bounds for A(n,2t+1). We present three practical results that prove the non-existence of perfect 2-error-correcting codes in Sn under the Hamming metric for numerous values of n. Specifically, we prove that 91 and 907 are the only values for n≤1000 for which Sn may contain a perfect 2-error-correcting code under the Hamming metric. Additionally, we prove that for any integer n such that n2-n+2 is divisible by a prime exceeding n-⌊n7⌋, (Formula presented.) The result improves the known upper bounds of A(n, 5) for all integers n≥35 such that n2-n+2 is divisible by a prime exceeding n-⌊n7⌋. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2025.
Publication Date: 2025
Iranian Journal Of Mathematical Sciences And Informatics (20089473)20(2)pp. 123-128
It is proved that if 1 + x + y or 1 + x − y cannot occur as a zero divisor of the complex group algebra of a finite group G for any two distinct x, y ∈ G \ {1}, then G is solvable. We also characterize all finite abelian groups with the latter property. The motivation of studying such property for finite groups is to settle the existence of zero divisors with support size 3 in the integral group algebra of torsion free residually finite groups. © 2025 Academic Center for Education, Culture and Research TMU.
Publication Date: 2024
Iranian Journal Of Mathematical Sciences And Informatics (20089473)19(2)pp. 189-194
It is proved that if ∑g∈Gagg is a non-zero zero divisor element of the complex group algebra ℂG of a torsion-free group G then2∑g∈G|ag|2 < (∑g∈G|ag|)2. © 2024 Academic Center for Education, Culture and Research TMU.
Publication Date: 2024
Siberian Electronic Mathematical Reports (18133304)21(2)pp. 645-653
In this paper we characterize non-abelian nite 2-gene-rator groups G whose non-commuting graphs are Aut(G)-symmetric. We also nd some general results on these groups. These partially answer Problem 31 posed in Peter Cameron’s home page, old problems. © 2024 Abdollahi A.
Publication Date: 2024
Communications in Algebra (00927872)52(8)pp. 3457-3460
A group G is called logically generated by a subset S, if every element of G can be defined by a first order formula with parameters from S. We consider the case where G is a direct product of finite nilpotent groups with mutually coprime orders and we show that logical and algebraic generations are equivalent in G. We also prove that in the case when G is a free non-abelian group, if S logically generates G then either it generates G algebraically or (Formula presented.) is not a free factor of G. © 2024 Taylor & Francis Group, LLC.
Publication Date: 2023
Communications in Algebra (00927872)51(3)pp. 1011-1019
A complex character χ of a finite group G is called sharp if (Formula presented.) where (Formula presented.). In this paper we give a characterization of finite groups all non-linear irreducible characters of which are sharp. © 2022 Taylor & Francis Group, LLC.
Publication Date: 2023
Cryptography and Communications (19362447)15(5)pp. 891-903
We give two methods that are based on the representation theory of symmetric groups to study the largest size P(n, d) of permutation codes of length n, i.e., subsets of the set Sn of all permutations on { 1 , ⋯ , n} with the minimum distance (at least) d under the Kendall τ -metric. The first method is an integer programming problem obtained from the transitive actions of Sn . The second method can be applied to refute the existence of perfect codes in Sn . Applying these methods, we reduce the known upper bound (n- 1) ! - 1 for P(n, 3) to (n-1)!-⌈n3⌉+2≤(n-1)!-2 , whenever n≥ 11 is prime. If n= 6 , 7, 11, 13, 14, 15, 17, the known upper bound for P(n, 3) is decreased by 3, 3, 9, 11, 1, 1, 4, respectively. © 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
Publication Date: 2023
Journal of Algebra (00218693)631pp. 136-147
Let k be any positive integer and G a compact (Hausdorff) group. Let npk(G) denote the probability that k+1 randomly chosen elements x1,…,xk+1 satisfy [x1,x2,…,xk+1]=1. We study the following problem: If npk(G)>0 then, does there exist an open nilpotent subgroup of class at most k? The answer is positive for profinite groups and we give a new proof. We also prove that the connected component G0 of G is abelian and there exists a closed normal nilpotent subgroup N of class at most k such that G0N is open in G. In particular, a connected compact group G with npk(G)>0 is abelian. © 2023 Elsevier Inc.
Publication Date: 2022
Communications in Algebra (00927872)50(6)pp. 2731-2739
Let χ be a virtual (generalized) character of a finite group G and (Formula presented.) be the image of χ on (Formula presented.) The pair (Formula presented.) is said to be sharp of type L or L-sharp if (Formula presented.) If the principal character of G is not an irreducible constituent of χ, the pair (Formula presented.) is called normalized. In this paper, we first provide some counterexamples to a conjecture that was proposed by Cameron and Kiyota in 1988. This conjecture states that if (Formula presented.) is L-sharp and (Formula presented.) then the inner product (Formula presented.) is uniquely determined by L. We then prove that this conjecture is true in the case that (Formula presented.) is normalized, χ is a character of G, and L contains at least an irrational value. © 2022 Taylor & Francis Group, LLC.
Publication Date: 2022
Bulletin of the Australian Mathematical Society (00049727)105(1)pp. 87-91
For any (Hausdorff) compact group G, denote by cp(G) the probability that a randomly chosen pair of elements of G commute. We prove that there exists a finite group H such that cp(G) = cp(H)/|G : F|2, where F is the FC-centre of G and H is isoclinic to F with cp(F) = cp(H) whenever cp(G) > 0. In addition, we prove that a compact group G with cp(G) > 3/40 is either solvable or isomorphic to A5 × Z(G), where A5 denotes the alternating group of degree five and the centre Z(G) of G contains the identity component of G. © 2021 Australian Mathematical Publishing Association Inc.
Publication Date: 2022
Mathematical Proceedings of the Cambridge Philosophical Society (03050041)173(2)pp. 329-332
The following question is proposed by Martino, Tointon, Valiunas and Ventura in [4, question 1·20]: Let G be a compact group, and suppose that Nk(G) = {(x1, ... , xk+1) ∈ Gk+1| [x1, ... , xk+1] = 1} has positive Haar measure in. Does G have an open k-step nilpotent subgroup? We give a positive answer for. © The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society.
Publication Date: 2022
Ars Mathematica Contemporanea (18553974)22(1)
Let G be a graph on n vertices and consider the adjacency spectrum of G as the ordered n-tuple whose entries are eigenvalues of G written decreasingly. Let G and H be two non-isomorphic graphs on n vertices with spectra S and T, respectively. Define the distance between the spectra of G and H as the distance of S and T to a norm N of the n-dimensional vector space over real numbers. Define the cospectrality of G as the minimum of distances between the spectrum of G and spectra of all other non-isomorphic n vertices graphs to the norm N. In this paper we investigate copsectralities of the cocktail party graph and the complete tripartite graph with parts of the same size to the Euclidean or Manhattan norms. © 2022 Society of Mathematicians, Physicists and Astronomers of Slovenia. All rights reserved.
Publication Date: 2022
Designs, Codes, and Cryptography (09251022)90(12)pp. 2841-2859
We study permutation codes which are groups and all of whose non-identity code elements have the same number of fixed points. It follows that over certain classes of groups such permutation codes exist. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
Publication Date: 2022
Bulletin Of The Iranian Mathematical Society (10186301)48(5)pp. 2079-2087
We show that there is no finite non-trivial p-group of coclass at most 4 except the dihedral group of order 8 which is isomorphic to its automorphism group. © 2021, Iranian Mathematical Society.
Publication Date: 2022
Bulletin Of The Iranian Mathematical Society (10186301)48(6)pp. 3809-3821
For a finite group G and its character χ, let Lχ be the image of χ on G- { 1 }. The pair (G, χ) is said to be sharp of type L if | G| = Π a∈L(χ(1) - a) , where L= Lχ. The pair (G, χ) is said to be normalized if the principal character of G is not an irreducible constituent of χ. In this paper, we study normalized sharp pairs of type L= { - 1 , 1 , 3 } proposed by Cameron and Kiyota in [J Algebra 115(1):125–143, 1988], under some additional hypotheses. © 2022, The Author(s) under exclusive licence to Iranian Mathematical Society.
In order to overcome the challenges posed by flash memories, the rank modulation scheme was proposed. In the rank modulation the codewords are permutations. In this paper, we study permutation codes with a specified length and minimum Kendall \tau-distance, and with as many codewords (permutations) as possible. We managed to make many significant improvements in the size of the best known codes. In particular, we show that for all n\geq 6 and for all \displaystyle \frac{3}{5}\begin{pmatrix}n\\2\end{pmatrix}\lt d\leq\frac{2}{3}\begin{pmatrix}n\\2\end{pmatrix} the largest size of a permutation code of length n and minimum distance at least d under Kendall \tau-metric is 4. © 2022 IEEE.
Publication Date: 2022
Comptes Rendus Mathematique (17783569)360pp. 1001-1008
An abstract group G is called totally 2-closed if H = H(2),Ω for any set Ω with G ∼= H ≤ Sym(Ω), where H(2),Ω is the largest subgroup of Sym(Ω) whose orbits on Ω ×Ω are the same orbits of H. In this paper, we classify the finite soluble totally 2-closed groups. We also prove that the Fitting subgroup of a totally 2-closed group is a totally 2-closed group. Finally, we prove that a finite insoluble totally 2-closed group G of minimal order with non-trivial Fitting subgroup has shape Z · X, with Z = Z(G) cyclic, and X is a finite group with a unique minimal normal subgroup, which is nonabelian. © 2022 Elsevier Masson SAS. All rights reserved.
Publication Date: 2022
Czechoslovak Mathematical Journal (00114642)72(4)pp. 1081-1087
For a complex character χ of a finite group G, it is known that the product sh(χ)=∏l∈L(χ)(χ(1)−l) is a multiple of ∣G∣, where L(χ) is the image of χ on G − {1} The character χ is said to be a sharp character of type L if L = L(χ) and sh(χ) = ∣G∣. If the principal character of G is not an irreducible constituent of χ, then the character χ is called normalized. It is proposed as a problem by P. J. Cameron and M. Kiyota, to find finite groups G with normalized sharp characters of type {−1, 0, 2}. Here we prove that such a group with nontrivial center is isomorphic to the dihedral group of order 12. © 2022, Institute of Mathematics, Czech Academy of Sciences.
Publication Date: 2022
Advances in Group Theory and Applications (24991287)13pp. 71-82
Lévai and Pyber [5] proposed the following as a conjecture (see also Problem 14.53 of [9]): if G is a profinite group such that the set of solutions of the equation xn = 1 has positive Haar measure, then G has an open subgroup H and an element t such that all elements of the coset tH have order dividing n. We define a constant cn for all finite groups and prove that the latter conjecture is equivalent with a conjecture saying cn < 1. Using the latter equivalence we observe that correctness of Lévai and Pyber conjecture implies the existence of the universal upper bound 1/1-cn on the index of generalized Hughes-Thompson subgroup Hn of finite groups whenever it is non-trivial. It is known that the latter is widely open even for all primes n = p > 5. For odd n we also prove that Lévai and Pyber conjecture is equivalent to show that cn is less than 1 whenever cn is only computed on finite solvable groups. The validity of the conjecture has been proved in [5] for n = 2. Here we confirm the conjecture for n = 3. © 2022 AGTA.
Publication Date: 2021
Discrete Mathematics, Algorithms and Applications (17938317)13(6)
We confirm the following conjecture which has been proposed by Stanić in [I. Jovanović and Z. Stanić, Spectral distances of graphs, Linear Algebra Appl. 436(5) (2012) 1425-1435]: 0.945≈limn→∞σ(Pn,Zn) =limn→∞σ(Wn,Zn) = 1 2limn→∞σ(Pn,Wn),limn→∞σ(C2n,Z2n) = 2, where σ(G1,G2) =∑i=1n|λ i(G1)-λi(G2)| is the spectral distance between n vertex non-isomorphic graphs G1 and G2 with adjacency spectra λ1(Gi) ≥ λ2(Gi) ≥⋯ ≥ λn(Gi) for i = 1, 2, and Pn and Cn denote the path and cycle on n vertices, respectively, Zn denotes the coalescence of Pn-2 and P3 on one of the vertices of degree 1 of Pn-2 and the vertex of degree 2 of P3, and Wn denotes the coalescence of Zn-2 and P3 on the vertex of degree 1 of Zn-2 which is adjacent to a vertex of degree 2 and the vertex of degree 2 of P3. © 2021 World Scientific Publishing Company.
Publication Date: 2021
Bulletin Of The Iranian Mathematical Society (10186301)47(6)pp. 1827-1848
The famous unit conjecture for group algebras states that every unit is trivial. The validity of this conjecture is not known for the sightly simple example of fours group Γ=⟨x,y|(x2)y=x-2,(y2)x=y-2⟩ which it is “the simplest” example of a torsion-free non unique-product supersoluble group. In this article for n∈ N, we set Hn=⟨x2n,y2n,(xy)2n⟩⩽Γ and we consider Gn= Γ / Hn. We will show that there is a large subset Nn of C[Gn] which its elements are non-unit, so all elements of the set N=⋃n∈Nφn-1(Nn) are non-unit in C[Γ] , where φn: C[Γ] → C[Gn] is the induced group ring homomorphism by the quotient map φn: Γ → Gn. © 2020, Iranian Mathematical Society.
Publication Date: 2021
Rendiconti del Seminario Matematico dell 'Universita' di Padova/Mathematical Journal of the University of Padova (22402926)145pp. 191-203
– A famous conjecture about group algebras of torsion-free groups states that there is no zero divisor in such group algebras. A recent approach to settle the conjecture is to show the non-existence of zero divisors with respect to the length of possible ones, where by the length we mean the size of the support of an element of the group algebra. The case length 2 cannot be happen. The first unsettled case is the existence of zero divisors of length 3. Here we study possible length 3 zero divisors in the rational group algebras and in the group algebras over the field Fp with p elements for some prime p. As a consequence we prove that the rational group algebras of torsion-free groups which are residually finite p-group for some prime p ≠ 3 have no zero divisor of length 3. We note that the determination of all zero divisors of length 3 in group algebras over F2 of cyclic groups is equivalent to find all trinomials (polynomials with 3 non-zero terms) divided by irreducible polynomials over F2 . The latter is a subject studied in coding theory and we add here some results, e.g. we show that 1 + x + x2 is a zero divisor in the group algebra over F2 for some element x of the group if and only if x is of finite order divided by 3 and we find all β in the group algebra of the shortest length such that (1+x +x2 )β = 0; and 1+x2 +x3 or 1+x +x3 is a zero divisor in the group algebra over F2 for some element x of the group if and only if x is of finite order divided by 7. © 2021 Università degli Studi di Padova.
Publication Date: 2020
Bulletin Of The Malaysian Mathematical Sciences Society (01266705)43(3)pp. 2313-2320
It is proved that automorphism groups of all 2-groups of coclass 2 are 2-groups, except only three ones; for 2-groups of coclass 3, there are only 20 groups and exactly six infinite sequences of 2-groups whose automorphism groups are not 2-groups. © 2019, Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia.
Publication Date: 2020
Journal of Algebra and its Applications (17936829)19(4)
Let G be a unique product group, i.e. for any two finite subsets A,B of G, there exists x G which can be uniquely expressed as a product of an element of A and an element of B. We prove that if C is a finite subset of G containing the identity element such that (C) is not abelian, then, for all subsets B of G with |B|≥ 7, |BC|≥|B| + |C| + 2. Also, we prove that if C is a finite subset containing the identity element of a torsion-free group G such that |C| = 3 and (C) is not abelian, then for all subsets B of G with |B|≥ 7, |BC|≥|B| + 5. Moreover, if (C) is not isomorphic to the Klein bottle group, i.e. the group with the presentation (x,y|xyx = y), then for all subsets B of G with |B|≥ 5, |BC|≥|B| + 5. The support of an element α =x Gαxx in a group algebra [G] ( is any field), denoted by supp(α), is the set {x G|αx0}. By the latter result, we prove that if αβ = 0 for some nonzero α,β [G] such that |supp(α)| = 3, then |supp(β)|≥ 12. Also, we prove that if αβ = 1 for some α,β [G] such that |supp(α)| = 3, then |supp(β)|≥ 10. These results improve a part of results in Schweitzer [J. Group Theory 16(5) (2013) 667-693] and Dykema et al. [Exp. Math. 24 (2015) 326-338] to arbitrary fields, respectively. © 2020 World Scientific Publishing Company.
Publication Date: 2020
Journal of Group Theory (14354446)23(6)pp. 991-998
Lévai and Pyber proposed the following as a conjecture: Let G be a profinite group such that the set of solutions of the equation x n = 1 {x^{n}=1} has positive Haar measure. Then G has an open subgroup H and an element t such that all elements of the coset tH have order dividing n (see [V. D. Mazurov and E. I. Khukhro, Unsolved Problems in Group Theory. The Kourovka Notebook. No. 19, Russian Academy of Sciences, Novosibirisk, 2019; Problem 14.53]). The validity of the conjecture has been proved in [L. Lévai and L. Pyber, Profinite groups with many commuting pairs or involutions, Arch. Math. (Basel) 75 2000, 1-7] for n = 2 {n=2}. Here we study the conjecture for compact groups G which are not necessarily profinite and n = 3 {n=3}; we show that in the latter case the group G contains an open normal 2-Engel subgroup. © 2020 Walter de Gruyter GmbH, Berlin/Boston 2020.
Publication Date: 2020
Journal of Algebra and its Applications (17936829)19(11)
We prove that a locally graded group whose proper subgroups are Engel (respectively, k-Engel) is either Engel (respectively, k-Engel) or finite. We also prove that a group of infinite rank whose proper subgroups of infinite rank are Engel (respectively, k-Engel) is itself Engel (respectively, k-Engel), provided that G belongs to the ernikov class , which is the closure of the class of periodic locally graded groups by the closure operations Ṕ, P`, R and L. © 2020 World Scientific Publishing Company.
Publication Date: 2020
Bulletin Of The Iranian Mathematical Society (10186301)46(5)pp. 1371-1387
We obtain some necessary conditions on coefficients of possible units of even (resp., odd) L-length in the group algebra KΓ , where Γ is the (Passman) fours group. In particular, the relations between the coefficients of possible units of L-length greater or equal to 4, in this group algebra, are also investigated. © 2019, Iranian Mathematical Society.
Publication Date: 2019
Designs, Codes, and Cryptography (09251022)87(10)pp. 2335-2340
Let M(n, d) be the maximum size of a permutation code of length n and distance d. In this note, the permutation codewords of a classical code C are considered. These are the codewords with all different entries in C. Using these codewords for Reed–Solomon codes, we present some good permutation codes in this class of codes. As a consequence, since these codes are subsets of Reed–Solomon codes, decoding algorithms known for Reed–Solomon codes can also be used as a decoding algorithm for them. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.
Publication Date: 2019
Journal of Algebra and its Applications (17936829)18(1)
Let G be a group and S an inverse closed subset of G\{1}. By a Cayley graph Cay(G,S), we mean the graph whose vertex set is the set of elements of G and two vertices x and y are adjacent if x-1y S. A group G is called a CI-group if Cay(G,S)≅Cay(G,T) for some inverse closed subsets S and T of G\{1}, then Sα = T for some automorphism α of G. A finite group G is called a BI-group if Cay(G,S)≅Cay(G,T) for some inverse closed subsets S and T of G\{1}, then M V S = M V T for all positive integers V, where M V S denotes the set {ΣsϵSX(s)|X(1) = V,X is a complex irreducible character of G}. It was asked by László Babai [Spectra of Cayley graphs, J. Combin. Theory Ser. B 27 (1979) 180-189] if every finite group is a BI-group; various examples of finite non-BI-groups are presented in [A. Abdollahi and M. Zallaghi, Character sums of Cayley graph, Comm. Algebra 43(12) (2015) 5159-5167]. It is noted in the latter paper that every finite CI-group is a BI-group and all abelian finite groups are BI-groups. However, it is known that there are finite abelian non-CI-groups. Existence of a finite non-abelian BI-group which is not a CI-group is the main question which we study here. We find two non-abelian BI-groups of orders 20 and 42 which are not CI-groups. We also list all BI-groups of orders up to 30. © 2019 World Scientific Publishing Company.
Publication Date: 2019
Journal of Algebra and its Applications (17936829)18(12)
The conjecture on units of group algebras of a torsion-free supersoluble group is saying that every unit is trivial, i.e. a product of a nonzero element of the field and an element of the group. This conjecture is still open and even in the slightly simple case of the fours group σ = (x,y|(x2)y = x-2, (y2)x = y-2), it is not yet known. The main result of this paper is to show that a wide range of elements of group algebra of σ are nonunit. © 2019 World Scientific Publishing Company.
Publication Date: 2019
Transactions On Combinatorics (22518657)8(1)pp. 15-40
Fixed-point-free permutations, also known as derangements, have been studied for centuries. In particular, depending on their applications, derangements of prime-power order and of prime order have always played a crucial role in a variety of different branches of mathematics: from number theory to algebraic graph theory. Substantial progress has been made on the study of derangements, many long- standing open problems have been solved, and many new research problems have arisen. The results obtained and the methods developed in this area have also effectively been used to solve other problems regarding finite vertex-transitive graphs. The methods used in this area range from deep group theory, including the classification of the finite simple groups, to combinatorial techniques. This article is devoted to surveying results, open problems and methods in this area. © 2019 University of Isfahan.
Publication Date: 2019
Communications in Algebra (00927872)47(1)pp. 424-449
Kaplansky Zero Divisor Conjecture states that if G is a torsion-free group and (Formula presented.) is a field, then the group ring (Formula presented.) contains no zero divisor and Kaplansky Unit Conjecture states that if G is a torsion-free group and (Formula presented.) is a field, then (Formula presented.) contains no non-trivial units. The support of an element (Formula presented.) in (Formula presented.), denoted by (Formula presented.), is the set (Formula presented.). In this paper, we study possible zero divisors and units with supports of size 4 in group algebras of torsion-free groups. We prove that if α, β are non-zero elements in (Formula presented.) for a possible torsion-free group G and an arbitrary field (Formula presented.) such that (Formula presented.) and (Formula presented.), then (Formula presented.). In [J. Group Theory, 16 (Formula presented.) no. 5, 667–693], it is proved that if (Formula presented.) is the field with two elements, G is a torsion-free group and (Formula presented.) such that (Formula presented.) and (Formula presented.), then (Formula presented.). We improve the latter result to (Formula presented.). Also, concerning the Unit Conjecture, we prove that if (Formula presented.) for some (Formula presented.) and (Formula presented.), then (Formula presented.). © 2019, © 2019 Taylor & Francis Group, LLC.
Publication Date: 2018
Journal of Algebra and its Applications (17936829)17(4)
Let G be any group and G be a subgroup of Sym(Ω) for some set Ω. The 2-closure of G on Ω, denoted by G(2),Ω, is by definition, {θ ∈ Sym(Ω) | ∀α, β ∈ Ω, ∃g ∈ G, αθ = αg, βθ = βg}. The group G is called 2-closed on Ω if G = G(2),Ω. We say that a group G is a totally 2-closed group if H = H(2),Ω for any set Ω such that G ≈H ≤ Sym(Ω). Here we show that the center of any finite totally 2-closed group is cyclic and a finite nilpotent group is totally 2-closed if and only if it is cyclic or a direct product of a generalized quaternion group with a cyclic group of odd order. © 2018 World Scientific Publishing Company.
Publication Date: 2018
Bulletin Of The Iranian Mathematical Society (10186301)44(4)pp. 1067-1068
Let G be a group. A set of proper subgroups of G is called a cover or covering for G if its set-theoretic union is equal to G. A cover for G is called irredundant if every proper subset of the cover is not again a cover for G. Yakov Berkovich proposed the following problem: does there exist a p-group G admitting an irredundant covering by n subgroups, where p+ 1 < n< 2 p? If ‘yes’, classify such groups. We prove that for any prime p≥ 3 , every finite p-group whose minimum number of generators is at least 3 has an irredundant cover of size 3(p+1)2. It follows that the classification of all finite p-groups having an irredundant covering of size n where p+ 1 < n< 2 p is not possible. © 2018, Iranian Mathematical Society.
Publication Date: 2018
Ars Combinatoria (03817032)140pp. 351-357
A graph is called integral whenever eigenvalues of its adjacency matrix are all integer. Connected graphs whose eigenvalues are distinct and main are called controllable graphs. In [Applicable Analysis and Discrete Mathematics, 5(2011), 165-175.] it is proved that the only controllable integral graph is the one-vertex graph K1. The proof of the latter result depends highly on the classification of integral graphs with at most 10 vertices, which in turn depends on computer computations. Here we give a machine-free proof for a slight generalization of the latter result: The only integral graphs whose eigenvalues are all distinct are K1, K2 and the disjoint union of K2 and K1. © 2018 Charles Babbage Research Centre. All rights reserved.
Publication Date: 2018
Communications in Algebra (00927872)46(2)pp. 887-925
Kaplansky’s zero divisor conjecture (unit conjecture, respectively) states that for a torsion-free group G and a field &, the group ring &[G] has no zero divisors (has no units with supports of size greater than 1). In this paper, we study possible zero divisors and units in &[G] whose supports have size 3. For any field & and all torsion-free groups G, we prove that if αβ = 0 for some non-zero α,β∈&[G] such that |supp(α)| = 3, then |supp(β)|≥10. If & = &2 is the field with 2 elements, the latter result can be improved so that |supp(β)|≥20. This improves a result in Schweitzer [J. Group Theory, 16 (2013), no. 5, 667–693]. Concerning the unit conjecture, we prove that if αβ = 1 for some α,β∈&[G] such that |supp(α)| = 3, then |supp(β)|≥9. The latter improves a part of a result in Dykema et al. [Exp. Math., 24 (2015), 326–338] to arbitrary fields. © 2018 Taylor & Francis.
Publication Date: 2017
Discrete Mathematics (0012365X)340(5)pp. 1116-1121
The adjacency spectrum of a graph Γ, which is denoted by Spec(Γ), is the multiset of eigenvalues of its adjacency matrix. We say that two graphs Γ and Γ′ are cospectral if Spec(Γ)=Spec(Γ′). In this paper for each prime number p, p≥23, we construct a large family of cospectral non-isomorphic Cayley graphs over the dihedral group of order 2p. © 2016 Elsevier B.V.
Publication Date: 2017
Designs, Codes, and Cryptography (09251022)84(1-2)pp. 73-85
We classify the distance-regular Cayley graphs with least eigenvalue - 2 and diameter at most three. Besides sporadic examples, these comprise of the lattice graphs, certain triangular graphs, and line graphs of incidence graphs of certain projective planes. In addition, we classify the possible connection sets for the lattice graphs and obtain some results on the structure of distance-regular Cayley line graphs of incidence graphs of generalized polygons. © 2016, The Author(s).
Publication Date: 2017
Bulletin Of The Malaysian Mathematical Sciences Society (01266705)40(4)pp. 1577-1589
A subgroup H of a group G is called a TI-subgroup if Hg∩ H= 1 or H for all g∈ G; and H is called quasi TI if CG(x) ≤ NG(H) for all non-trivial elements x∈ H. A group G is called (quasi CTI-group) CTI-group if every cyclic subgroup of G is a (quasi TI-subgroup) TI-subgroup. It is clear that TI subgroups are quasi TI. We first show that finite nilpotent quasi CTI-groups are CTI. In this paper, we classify all finite nilpotent CTI-groups. © 2015, Malaysian Mathematical Sciences Society and Universiti Sains Malaysia.
Publication Date: 2017
Journal of Algebra and its Applications (17936829)16(6)
In this paper we study finite p-groups G for which the p-part of |Aut(G): Inn(G)| has the least possible value p. We characterize the groups in some special cases, including p-groups of nilpotency class 2, of maximal class, of order at most p5, with cyclic Frattini subgroup and p-groups G in which |G: Z(G)|≤ p4. © 2017 World Scientific Publishing Company.
Publication Date: 2017
Electronic Journal of Combinatorics (10778926)24(3)
We show that the subgroup lattice of any finite group satisfies Frankl’s Union-Closed Conjecture. We show the same for all lattices with a modular coatom, a family which includes all supersolvable and dually semimodular lattices. A common technical result used to prove both may be of some independent interest. © 2017, Australian National University. All rights reserved.
Publication Date: 2017
International Journal of Algebra and Computation (02181967)27(7)pp. 849-862
A group G satisfies a positive generalized identity of degree n if there exist elements g1gn G such that xgn=1 for all x G. The minimum degree of such an identity is called the generalized exponent of G. Among other things, we prove that every finitely generated solvable group satisfying a positive generalized identity of prime degree is a finite p-group. Consequently, we show that every finite group with a positive generalized identity of degree 5 is a 5-group of exponent dividing 25. © 2017 World Scientific Publishing Company.
Publication Date: 2017
Communications in Algebra (00927872)45(12)pp. 5188-5192
We prove that the Tate cohomology groups Ĥn(G∕Φ(G),Z(Φ(G))) are non-trivial, whenever G is a finite p-group of class 3, or the pth term of the upper central series of G contains Z(Φ(G)). This confirms a conjecture of Schmid for these groups. © 2017 Taylor & Francis.
Publication Date: 2017
Communications in Algebra (00927872)45(8)pp. 3636-3642
A longstanding conjecture asserts that every finite nonabelian p-group admits a noninner automorphism of order p. In this paper we give some necessary conditions for a possible counterexample G to this conjecture, in the case when G is a 2-generator finite p-group. Then we show that every 2-generator finite p-group with abelian Frattini subgroup has a noninner automorphism of order p. © 2017 Taylor & Francis.
Publication Date: 2016
Ars Combinatoria (03817032)126pp. 301-310
Let ∗ be a binary graph operation. We call ∗ a Cayley operation if γ1 ∗ γ2 is a Cayley graph for any two Cayley graphs γ1 and γ2- In this paper, we prove that the cartesian, (categorical or tensor) direct and lexicographic products are Cayley operations. We also investigate the following question: Under what conditions on a binary graph operation ∗ and Cayley graphs γ1 ∗ γ2, the graph product γ1 ∗ γ2 is again a Cayley graph. The latter question is studied for the union, join (sum), replacement and zig-zag products of graphs.
Publication Date: 2016
Journal of Algebra and its Applications (17936829)15(9)
The adjacency spectrum Spec(Γ) of a graph Γ is the multiset of eigenvalues of its adjacency matrix. Two graphs with the same spectrum are called cospectral. A graph Γ is "determined by its spectrum" (DS for short) if every graph cospectral to it is in fact isomorphic to it. A group is DS if all of its Cayley graphs are DS. A group G is Cay-DS if every two cospectral Cayley graphs of G are isomorphic. In this paper, we study finite DS groups and finite Cay-DS groups. In particular we prove that a finite DS group is solvable, and every non-cyclic Sylow subgroup of a finite DS group is of order 4, 8, 16 or 9. We also give several infinite families of non-Cay-DS solvable groups. In particular we prove that there exist two cospectral non-isomorphic 6-regular Cayley graphs on the dihedral group of order 2p for any prime p ≥ 13. © 2016 World Scientific Publishing Company.
Publication Date: 2015
Communications in Algebra (00927872)43(12)pp. 5159-5167
Following [1], by a Cayley digraph we mean a graph Cay(G, S) whose vertex set is a group G, and there exists a directed edge from a vertex g to another vertex h if g-1h ∈ S, where S is a generating subset of G. The graph Cay (G, S) is called a Cayley graph if S = S-1 and 1 ∉ S. In Problem 3.3 of the above cited article, the following question is proposed. Let G be a finite group, let Γ = Cay(G, S) be a Cayley digraph, v a positive integer, and (Formula Presented.) It is easy to see that the set Mv is an invariant for the Cayley digraphs if the underlying group G is abelian. Here we negatively answer the above problem. We show that for every n ≥ 4 there is a Cayley graph Γn on the symmetric group Sn so that the above set Mv is not an invariant of Γn. We also find some other groups with the latter property. © 2015, Copyright Taylor & Francis Group, LLC.
Publication Date: 2015
Linear Algebra and Its Applications (00243795)466pp. 401-408
Richard Brualdi proposed in Stevanivić (2007) [10] the following problem: (Problem AWGS.4) Let Gn and Gn′ be two nonisomorphic graphs on n vertices with spectraλ1≥λ2≥âi»≥λnandλ1′≥λ2′≥âi»≥λn′, respectively. Define the distance between the spectra of Gn and Gn′ asλ(Gn,Gn′)=λi=1n(λi(or use λi=1n|λi-λi′|). Define the cospectrality of Gn bycs(Gn)=min{λ(Gn,Gn′):Gn′ not isomorphic to Gn}. Letcsn=max{cs(Gn):Gn a graph on n vertices}. Investigate cs(Gn) for special classes of graphs. Find a good upper bound on csn. In this paper we completely answer Problem B by proving that csn=2 for all n≥2, whenever csn is computed with respect to any â.,"p-norm with 1lt∞ and ;bsupesup=1 with respect to the ;bsupâ.,"∞;esup-norm. The cospectrality cs(Km,;bsubesub) of the complete bipartite graph Km,;bsubesub for all positive integers m and n with m≤n
Publication Date: 2015
Journal of Group Theory (14354446)18(1)pp. 111-114
The following problem was proposed as Problem 18.57 in [4] by D.V. Lytkina: Let G be a finite 2-group generated by involutions in which [x, u, u] = 1 for every x ∈ G and every involution u ∈ G. Is the derived length of G bounded? The question asks for an upper bound on the derived length of finite 2-groups generated by involutions in which every involution (not only the generators) is left 2-Engel. We negatively answer the question. © de Gruyter 2015.
Publication Date: 2014
Linear Algebra and Its Applications (00243795)451pp. 169-181
Richard Brualdi proposed in Stevanivić (2007) [6] the following problem: (Problem AWGS.4) Let Gn and Gn′ be two nonisomorphic graphs on n vertices with spectra λ1≥ λ 2≥⋯ ≥ λn and λ1′≥ λ2′≥⋯≥λn′, respectively. Define the distance between the spectra of Gn and Gn′ asλ(G n,Gn′)=i=1n (λi- λi′)2(or use i=1n|λ i-λi′|). Define the cospectrality of Gn bycs(Gn)=min{λ( Gn,Gn′):Gn′ not isomorphic to Gn}. Letcsn=max{cs(Gn):G n a graph on n vertices}. Investigate cs(Gn) for special classes of graphs. Find a good upper bound on csn. In this paper we study Problem A and determine the cospectrality of certain graphs by the Euclidian distance. Let Kn denote the complete graph on n vertices, nK1 denote the null graph on n vertices and K2+(n-2) K1 denote the disjoint union of the K2 with n-2 isolated vertices, where n≥2. In this paper we find cs(Kn), cs(n K1), cs(K2+(n-2)K1) (n≥2) and cs(Kn, n).© 2014 PublishedbyElsevierInc.
Publication Date: 2014
European Journal of Combinatorics (10959971)38pp. 102-109
Let G be a finite group, S ⊆ G {1} be a set such that if a ∈ S, then a -1 ∈ S, where 1 denotes the identity element of G. The undirected Cayley graph C a y (G, S) of G over the set S is the graph whose vertex set is G and two vertices a and b are adjacent whenever ab -1 ∈ S. The adjacency spectrum of a graph is the multiset of all eigenvalues of the adjacency matrix of the graph. A graph is called integral whenever all adjacency spectrum elements are integers. Following Klotz and Sander, we call a group G Cayley integral whenever all undirected Cayley graphs over G are integral. Finite abelian Cayley integral groups are classified by Klotz and Sander as finite abelian groups of exponent dividing 4 or 6. Klotz and Sander have proposed the determination of all non-abelian Cayley integral groups. In this paper we complete the classification of finite Cayley integral groups by proving that finite non-abelian Cayley integral groups are the symmetric group S3 of degree 3, C3 ⋊ C4 and Q8×C2n for some integer n ≥ 0, where Q8 is the quaternion group of order 8. © 2013 Elsevier Ltd.
Publication Date: 2014
Ars Combinatoria (03817032)113pp. 337-339
Let n be a positive integer. Denote by PG(n,q) the n-dimensional projective space over the finite field Fq, of order q. A blocking set in PG(n,q) is a set of points that has non-empty intersection with every hyperplane of PG(n,q). A blocking set is called minimal if none of its proper subsets are blocking sets. In this note we prove that if PG(n,q) contains a minimal blocking set of size κi for i ∈ {1,2}, then PG(n1 + n2 + 1,q) contains a minimal blocking set of size κ1+κ2 - 1. This result is proved by a result on groups with maximal irredundant covers.
Publication Date: 2014
Communications in Algebra (00927872)42(9)pp. 3944-3949
Let G be a non-abelian group and Z(G) be the center of G. The non-commuting graph ΓG associated to G is the graph whose vertex set is G{set minus}Z(G) and two distinct elements x, y are adjacent if and only if xy ≠ yx. We prove that if G and H are non-abelian nilpotent groups with irregular isomorphic non-commuting graphs, then {pipe}G{pipe} = {pipe}H{pipe}. © 2014 Copyright Taylor & Francis Group, LLC.
Publication Date: 2014
Journal of Group Theory (14354446)17(2)pp. 267-272
Every finite p-group of coclass 2 has a noninner automorphism of order p leaving the center elementwise fixed. © 2014 de Gruyter.
Publication Date: 2014
Bulletin of the Australian Mathematical Society (00049727)90(2)pp. 227-231
Let G be a finite 2-group. If G is of coclass 2 or (G, Z(G)) is a Camina pair, then G admits a noninner automorphism of order 2 or 4 leaving the Frattini subgroup elementwise fixed. © 2014 Australian Mathematical Publishing Association Inc.
Publication Date: 2014
Glasnik Matematicki (0017095X)49(1)pp. 119-122
An automorphism of a group is called outer if it is not an inner automorphism. Let G be a finite p-group. Then for every outer p-automorphism Φ of G the subgroup CG(Φ) = {x∈G | xΦ = x} has order p if and only if G is of order at most p2.
Publication Date: 2013
Communications in Algebra (00927872)41(2)pp. 451-461
Let G be a non-abelian group and Z(G) be the center of G. Associate a graph ΓG (called noncommuting graph of G) with G as follows: Take G{set minus}Z(G) as the vertices of ΓG, and join two distinct vertices x and y, whenever xy ≠ yx. Here, we prove that "the commutativity pattern of a finite non-abelian p-group determine its order among the class of groups"; this means that if P is a finite non-abelian p-group such that ΓP ≅ ΓH for some group H, then {pipe}P{pipe} = {pipe}H{pipe}. © 2013 Copyright Taylor and Francis Group, LLC.
Publication Date: 2013
Beitrage zur Algebra und Geometrie (01384821)54(1)pp. 363-381
Let p be a prime. We prove that a finite p-group of class 3 has a noninner automorphism of order p. A result counting derivations from an abelian p-group to an elementary abelian one is of independent interest. © 2012 The Managing Editors.
Publication Date: 2013
International Journal Of Group Theory (22517669)2(4)pp. 17-20
A longstanding conjecture asserts that every finite nonabelian p-group admits a noninner automorphism of order p. Let G be a finite nonabelian p-group. It is known that if G is regular or of nilpotency class 2 or the commutator subgroup of G is cyclic, or G/Z(G) is powerful, then G has a noninner automorphism of order p leaving either the center Z(G) or the Frattini subgroup Φ(G) of G elementwise fixed. In this note, we prove that the latter noninner automorphism can be chosen so that it leaves Z(G) elementwise fixed. © 2013 University of Isfahan.
Publication Date: 2013
Applicable Analysis and Discrete Mathematics (14528630)7(1)pp. 119-128
Let G be a non-trivial finite group, S ⊇ G\{e} be a set such that if a ∈ S, then a-1 ∈ S and e be the identity element of G. Suppose that Cay(G, S) is the Cayley graph with the vertex set G such that two vertices a and b are adjacent whenever ab-1 ∈ S. An arbitrary graph is called integral whenever all eigenvalues of the adjacency matrix are integers. We say that a group G is Cayley integral simple whenever every connected integral Cayley graph on G is isomorphic to a complete multipartite graph. In this paper we prove that if G is a non-simple group, then G is Cayley integral simple if and only if G ≅ ℤp2 for some prime number p or G ≅ ℤ2×ℤ2. Moreover, we show that there exist finite non-abelian simple groups which are not Cayley integral simple.
Publication Date: 2013
Filomat (03545180)27(1)pp. 57-63
Let G be an (n,m)-graph (n vertices and m-regular) and H be an (m, d)-graph. Randomly number the edges around each vertex of G by {1,...,m} and fix it. Then the replacement product GrH of graphs G®H (with respect to the numbering) has vertex set V(G®H) = V(G) × V(H) and there is an edge between (v, k) and (w, l) if v = w and kl ∈ E(H) or vw ∈ E(G) and kth edge incident on vertex v in G is connected to the vertex w and this edge is the lth edge incident on w in G, where the numberings k and l refers to the random numberings of edges adjacent to any vertex of G. If the set of edges of a graph can be partitioned to a set of complete matchings, then the graph is called 1-factorizable and any such partition is called a 1-factorization. In this paper, 1-factorizability of the replacement product GrH of graphs G®H is studied. As an application we show that fullerene C60 and C4C8 nanotorus are 1-factorizable.
Publication Date: 2012
Journal of Algebra (00218693)357pp. 203-207
Let G be any non-abelian group and Z(G) be its center. The non-commuting graph Γ G of G is the simple graph whose vertex set is G\Z(G), with two vertices x and y adjacent whenever xy≠xy. We prove that if Γ G is isomorphic to the non-commuting graph of the alternating group A n (n≥4), then G≅A n. This result together with a recent one due to Solomon and Woldar gives a complete positive answer to a conjecture proposed in [A. Abdollahi, S. Akbari, H.R. Maimani, Non-commuting graph of a group, J. Algebra 298 (2006) 468-492]: If S is any finite non-abelian simple group such that Γ S≅Γ G for some group G, then G≅S. © 2012 Elsevier Inc.
Publication Date: 2012
Bulletin Of The Iranian Mathematical Society (1017060X)38(2)pp. 305-311
Let R be an infinite ring. Here, we prove that if 0R be- longs to {x1x2 · · · xn | x1,x2,...,xn ε X} for every infinite subset X of R, then R satisfies the polynomial identity xn = 0. Also, we prove that if 0r belongs to {x1x2 · · · xn - xn+1 | x1, x2,..., xn, xn+1 ε X} for every infinite subset X of R, then xn = x, for all x ε R. © 2012 Iranian Mathematical Society.
Publication Date: 2011
Journal of Algebra (00218693)342(1)pp. 154-160
We show the existence of cohomologically trivial Q-module A, where Q=G/Φ(G), A=Z(Φ(G)), G is a finite non-abelian p-group, Φ(G) is the Frattini subgroup of G, Z(Φ(G)) is the center of Φ(G), and Q acts on A by conjugation, i.e., zgΦ(G):=zg=gΦ1zg for all gΦG and all zΦZ(Φ(G)). This means that the Tate cohomology groups Hn(Q,A) are all trivial for any nΦZ. Our main result answers Problem 17.2 of [V.D. Mazurov, E.I. Khukhro (Eds.), The Kourovka Notebook. Unsolved Problems in Group Theory, seventeenth edition, Russian Academy of Sciences, Siberian Division, Institute of Mathematics, Novosibirsk, 2010] proposed by P. Schmid. © 2011 Elsevier Inc.
Publication Date: 2011
Journal of Algebra (00218693)347(1)pp. 53-59
A subset S of a group G is called an Engel set if, for all x, y∈. S, there is a non-negative integer n=n(x, y) such that [x,ny]=1. In this paper we are interested in finding conditions for a group generated by a finite Engel set to be nilpotent. In particular, we focus our investigation on groups generated by an Engel set of size two. © 2011 Elsevier Inc.
Publication Date: 2011
Electronic Journal of Combinatorics (10778926)18(1)
A graph is called integral, if its adjacency eigenvalues are integers. In this paper we determine integral quartic Cayley graphs on finite abelian groups. As a side result we show that there are exactly 27 connected integral Cayley graphs up to 11 vertices.
Publication Date: 2011
Archiv der Mathematik (0003889X)97(5)pp. 407-412
Let θ be a word in n variables and let G be a group; the marginal and verbal subgroups of G determined by θ are denoted by θ(G) and θ*(G), respectively. The following problems are generally attributed to P. Hall: (I)If π is a set of primes and {pipe}G: θ*(G){pipe} is a finite π-group, is θ(G) also a finite π-group? (II)If θ(G) is finite and G satisfies maximal condition on its subgroups, is {pipe}G: θ*(G){pipe} finite? (III)If the set {θ(g1,...,gn) {pipe} g1,...,gn ∈ G} is finite, does it follow that θ(G) is finite? We investigate the case in which θ is the n-Engel word en = [x,n y] for n∈ {2,3,4}. © 2011 Springer Basel AG.
Publication Date: 2010
Communications in Algebra (00927872)38(12)pp. 4390-4403
We associate a graph N G with a group G (called the non-nilpotent graph of G) as follows: take G as the vertex set and two vertices are adjacent if they generate a non-nilpotent subgroup. In this article, we study the graph theoretical properties of N G and its induced subgraph on G \ nil(G), where nil(G) = {x ε G {pipe}(x, y)is nilpotent for all y ε G}. For any finite group G, we prove that N G has either {pipe} Z.ast;(G){pipe} or {pipe}Z.ast;(G){pipe} +1 connected components, where Z.ast;(G) is the hypercenter of G. We give a new characterization for finite nilpotent groups in terms of the non-nilpotent graph. In fact, we prove that a finite group G is nilpotent if and only if the set of vertex degrees of N G has at most two elements. © 2010 Copyright Taylor and Francis Group, LLC.
Publication Date: 2010
Algebra Colloquium (02191733)17(4)pp. 611-620
Let G be a non-abelian group. The non-commuting graph AG of G is defined as the graph whose vertex set is the non-central elements of G and two vertices are joint if and only if they do not commute. In a finite simple graph Γ, the maximum size of complete subgraphs of Γ is called the clique number of Γ and denoted by ω(Γ). In this paper, we characterize all non-solvable groups G with ω(AG)≤ 57, where 57 is the clique number of the non-commuting graph of the projective special linear group PSL(2,7). We also determine ω(AG) for all finite minimal simple groups G. © 2010 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and Suzhou University.
Publication Date: 2010
Communications in Algebra (00927872)38(3)pp. 933-943
In this article we study left and right 4-Engel elements of a group. In particular, we prove that (a ab) is nilpotent of class at most 4, whenever a is of finite order and b±1 are right 4-Engel elements or a±1 are left 4-Engel elements and b is an arbitrary element of G. Furthermore, we prove that for any prime p and any element a of finite p-power order in a group G such that a±1 ∈ L4(G), a4, if p = 2, and ap, if p is an odd prime number, is in the Baer radical of G. © Taylor & Francis Group, LLC.
Publication Date: 2010
Journal of Algebra (00218693)323(3)pp. 779-789
In this paper we study the longstanding conjecture of whether there exists a non-inner automorphism of order p for a finite non-abelian p-group. We prove that if G is a finite non-abelian p-group such that G / Z (G) is powerful then G has a non-inner automorphism of order p leaving either Φ (G) or Ω1 (Z (G)) elementwise fixed. We also recall a connection between the conjecture and a cohomological problem and we give an alternative proof of the latter result for odd p, by showing that the Tate cohomology Hn (G / N, Z (N)) ≠ 0 for all n ≥ 0, where G is a finite p-group, p is odd, G / Z (G) is p-central (i.e., elements of order p are central) and N◁G with G / N non-cyclic. © 2009 Elsevier Inc. All rights reserved.
Publication Date: 2010
Journal of Algebra and its Applications (17936829)9(5)pp. 763-769
We prove that the set of right 4-Engel elements of a group G is a subgroup for locally nilpotent groups G without elements of orders 2, 3 or 5; and in this case the normal closure xG is nilpotent of class at most 7 for each right 4-Engel elements x of G. © 2010 World Scientific Publishing Company.
Publication Date: 2009
Beitrage zur Algebra und Geometrie (01384821)50(2)pp. 443-448
Let G be a non-abelian group and Z(G) be its center. The non-commuting graph AG of G is the graph whose vertex set is G\Z(G) and two vertices are joined by an edge if they do not commute. Let SL(2,q) be the special linear group of degree 2 over the finite field of order q. In this paper we prove that if G is a group such that AG ≅ ASL(2,q) for some prime power q ≥ 2, then G ≅ SL(2,q). © 2009 Heldermann Verlag.
Publication Date: 2009
Journal of Algebra and its Applications (17936829)8(3)pp. 339-350
The concept of configuration was first introduced by Rosenblatt and Willis to give a condition for amenability of groups. We show that if G1 and G2 have the same configuration sets and H1 is a normal subgroup of G1 with abelian quotient, then there is a normal subgroup H2 of G2 such that G1/H1 ≅ G2/H2. Also configuration of FC-groups and isomorphism is studied. © World Scientific Publishing Company.
Publication Date: 2009
Journal of Algebra and its Applications (17936829)8(2)pp. 243-257
We associate a graph CG to a non locally cyclic group G (called the non-cyclic graph of G) as follows: take G\Cyc (G) as vertex set, where Cyc (G) = {x ∈ G 〈x, y〉 is cyclic for all y ∈ G} is called the cyclicizer of G, and join two vertices if they do not generate a cyclic subgroup. For a simple graph Γ, w(Γ) denotes the clique number of Γ, which is the maximum size (if it exists) of a complete subgraph of Γ. In this paper we characterize groups whose non-cyclic graphs have clique numbers at most 4. We prove that a non-cyclic group G is solvable whenever w(CG) < 31 and the equality for a non-solvable group G holds if and only if G/Cyc(G) ≅ A5 or S5. © World Scientific Publishing Company.
Publication Date: 2009
Utilitas Mathematica (03153681)78pp. 243-250
In this paper we study the automorphism group of a possible symmetric (81, 16, 3) design.
Publication Date: 2009
Electronic Journal of Combinatorics (10778926)16(1)
Let G be a non-trivial group, S⊆G \ {1} and S = S-1 := {s-1 | s⊆ S}. The Cayley graph of G denoted by Γ(S : G) is a graph with vertex set G and two vertices a and b are adjacent if ab -1 ⊆ S. A graph is called integral, if its adjacency eigenvalues are integers. In this paper we determine all connected cubic integral Cayley graphs. We also introduce some infinite families of connected integral Cayley graphs.
Publication Date: 2008
Ars Combinatoria (03817032)86pp. 129-131
In this note we prove that all connected Cayley graphs of every finite group Q×H are 1-factorizable, where Q is any non-trivial group of 2-power order and H is any group of odd order.
Publication Date: 2008
Communications in Algebra (00927872)36(10)pp. 3783-3791
A group in which every element commutes with its endomorphic images is called an E-group. Our main result is that all 3-generator E-groups are abelian. It follows that the minimal number of generators of a finitely generated non-abelian E-group is four. Copyright © Taylor & Francis Group, LLC.
Publication Date: 2008
Ars Combinatoria (03817032)86pp. 409-413
We introduce a new technique for constructing pairwise balanced designs and group divisible designs from finite groups. These constructed designs do not give designs with new parameters but our construction gives rise to designs having a transitive automorphism group that also preserves the resolution classes.
Publication Date: 2008
Linear Algebra and Its Applications (00243795)428(11-12)pp. 2947-2954
Let R be a non-commutative ring and Z (R) be its center. The commuting graph of R is defined to be the graph Γ (R) whose vertex set is R {minus 45 degree rule} Z (R) and two distinct vertices are joint by an edge whenever they commute. Let F be a finite field, n ≥ 2 an arbitrary integer and R be a ring with identity such that Γ (R) ≅ Γ (Mn (F)), where Mn (F) is the ring of n × n matrices over F. Here we prove that | R | = | Mn (F) |. We also show that if | F | is prime and n = 2, then R ≅ M2 (F). © 2008 Elsevier Inc. All rights reserved.
Publication Date: 2008
Communications in Algebra (00927872)36(2)pp. 365-380
In this article we prove that a set of points B of PG(n, 2) is a minimal blocking set if and only if B=PG(d, 2) with d odd and B is a set of d+2 points of PG(d, 2) no d+1 of them in the same hyperplane. As a corollary to the latter result we show that if G is a finite 2-group and n is a positive integer, then G admits a n+1-cover if and only if n is even and G (C2)n, where by a m-cover for a group H we mean a set C of size m of maximal subgroups of H whose set-theoretic union is the whole H and no proper subset of C has the latter property and the intersection of the maximal subgroups is core-free. Also for all n10 we find all pairs (m,p) (m0 an integer and p a prime number) for which there is a blocking set B of size n in PG(m,p) such that B=PG(m,p).
Publication Date: 2008
Publicationes Mathematicae Debrecen (00333883)72(1-2)pp. 167-172
Let G be a group that is a set-theortic union of finitely many proper subgroups. Cohn defined σ(G) to be the least integer m such that G is the union of m proper subgroups. Determining σ is an open problem for most non-solvable groups. In this paper we give a formula for σ(G), where G is a completely reducible group.
Publication Date: 2008
Communications in Algebra (00927872)36(5)pp. 1976-1987
A group in which every element commutes with its endomorphic images is called an "E-group". If p is a prime number, a p-group G which is an E-group is called a "pE-group". Every abelian group is obviously an E-group. We prove that every 2-generator E-group is abelian and that all 3-generator E-groups are nilpotent of class at most 2. It is also proved that every infinite 3-generator E-group is abelian. We conjecture that every finite 3-generator E-group should be abelian. Moreover, we show that the minimum order of a non-abelian pE-group is p8 for any odd prime number p and this order is 27 for p = 2. Some of these results are proved for a class wider than the class of E-groups. Copyright © Taylor & Francis Group, LLC.
Publication Date: 2007
Journal of Algebra (00218693)318(2)pp. 680-691
Let G be a non-Engel group and let L (G) be the set of all left Engel elements of G. Associate with G a graph EG as follows: Take G \ L (G) as vertices of EG and join two distinct vertices x and y whenever [x,k y] ≠ 1 and [y,k x] ≠ 1 for all positive integers k. We call EG, the Engel graph of G. In this paper we study the graph theoretical properties of EG. © 2007 Elsevier Inc. All rights reserved.
Publication Date: 2007
Journal of Algebra (00218693)312(2)pp. 876-879
We prove that for any prime number p, every finite non-abelian p-group G of class 2 has a noninner automorphism of order p leaving either the Frattini subgroup Φ (G) or Ω1 (Z (G)) elementwise fixed. © 2006 Elsevier Inc. All rights reserved.
Publication Date: 2007
Houston Journal of Mathematics (03621588)33(1)pp. 43-57
Let G be a. group and let cent(G) denote the set of centralizers of single elements of G. A group G is called n-centralizer if |eerat(G)| = n. In this paper, for a finite group G, we give some interesting relations between \cent(G)\ and the maximum number of the pairwise non-commuting elements in G. Also we characterize all n-centralizer finite groups for n = 7 and 8. Using these results we prove that there is no finite group G with the property that \cent(G)\ = \cent(G/Z(G))\ = 8, where Z(G) denotes the centre of G. This latter result answers positively a conjecture posed by A. R. Ashrafi. © 2007 University of Houston.
Publication Date: 2007
Communications in Algebra (00927872)35(7)pp. 2057-2081
We associate a graph G to a nonlocally cyclic group G (called the noncyclic graph of G) as follows: take G\ Cyc(G) as vertex set, where Cyc(G)={xG| x, y is cyclic for all yG}, and join two vertices if they do not generate a cyclic subgroup. We study the properties of this graph and we establish some graph theoretical properties (such as regularity) of this graph in terms of the group ones. We prove that the clique number of G is finite if and only if G has no infinite clique. We prove that if G is a finite nilpotent group and H is a group with GH and |Cyc(G)|=|Cyc(H)|=1, then H is a finite nilpotent group. We give some examples of groups G whose noncyclic graphs are "unique", i.e., if GH for some group H, then GH. In view of these examples, we conjecture that every finite nonabelian simple group has a unique noncyclic graph. Also we give some examples of finite noncyclic groups G with the property that if GH for some group H, then |G|=|H|. These suggest the question whether the latter property holds for all finite noncyclic groups.
Publication Date: 2007
Communications in Algebra (00927872)35(4)pp. 1323-1332
We show that if R is an infinite ring such that XYYX for all infinite subsets X and Y, then R is commutative. We also prove that in an infinite ring R, an element aR is central if and only if aXXa for all infinite subsets X.
Publication Date: 2007
Journal of Pure and Applied Algebra (00224049)209(2)pp. 291-300
A cover for a group is a collection of proper subgroups whose union is the whole group. A cover is irredundant if no proper sub-collection is also a cover and is called maximal if all its members are maximal subgroups. For an integer n > 2, a cover with n members is called an n-cover. In this paper we determine groups with a maximal irredundant 7-cover with core-free intersection. The intersection of an irredundant n-cover is known to have index bounded by a function of n, though in general the precise bound is not known. Here we prove that the exact bound is 81 when n is 7. © 2006 Elsevier Ltd. All rights reserved.
Publication Date: 2006
Bulletin of the Australian Mathematical Society (00049727)74(1-5)pp. 121-132
This paper is an attempt to provide a partial answer to the following question put forward by Bernhard H. Neumann in 2000: "Let G be a finite group of order g and assume that however a set M of m elements and a set N of n elements of the group is chosen, at least one element of M commutes with at least one element of N. What relations between g, m, n guarantee that G is Abelian?" We find an exponential function f(m, n) such that every such group G is Abelian whenever |G| > f(m, n) and this function can be taken to be polynomial if G is not soluble. We give an upper bound in terms of m and n for the solubility length of G, if G is soluble. Copyright Clearance Centre, Inc.
Publication Date: 2006
BULLETIN OF THE BELGIAN MATHEMATICAL SOCIETY-SIMON STEVIN (13701444)13(2)pp. 287-294
Let n be an integer >= 2. A group G is called generalized n-abelian if it admits a positive polynomial endomorphism of degree n, that is if these exist n elements a(1), a(2),..., a(n) of G such that the function phi : x -> x(a1)x(a2) ... x(an) is an endomorphism of G. In this paper we give some sufficient conditions for a generalized n-abelian group to be abelian. In particular, we show that every group admitting a positive polynomial monomorphism of degree 3 is abelian.
Publication Date: 2006
Journal of Algebra (00218693)298(2)pp. 468-492
Let G be a non-abelian group and let Z ( G ) be the center of G. Associate a graph ΓG (called non-commuting graph of G) with G as follows: Take G \ Z ( G ) as the vertices of ΓG and join two distinct vertices x and y, whenever x y ≠ y x. We want to explore how the graph theoretical properties of ΓG can effect on the group theoretical properties of G. We conjecture that if G and H are two non-abelian finite groups such that ΓG ≅ ΓH, then | G | = | H |. Among other results we show that if G is a finite non-abelian nilpotent group and H is a group such that ΓG ≅ ΓH and | G | = | H |, then H is nilpotent. © 2006 Elsevier Inc. All rights reserved.
Publication Date: 2005
Communications in Algebra (00927872)33(5)pp. 1417-1425
Let n be an integer greater than 1. A group G is said to be n-rewritable (or a Qn-group) if for every n elements x1, x 2,...,xn in G there exist distinct permutations σ and τ in Sn such that xσ(1)x σ(2)⋯xσ(n)=xτ(1)x τ(2)⋯xτ(n). In this paper, we characterize all 3-rewritable nilpotent 2-groups of class 2. Also we have found a bound for the nilpotency class of certain nilpotent 3-rewritable groups, and have shown that 3-rewritable groups satisfy a certain law. Copyright © Taylor & Francis, Inc.
Publication Date: 2005
Journal of Algebra (00218693)283(2)pp. 431-446
Let n be a positive integer. We say that a group G satisfies the condition ε(n), if every set of n+1 elements of G contains a pair {x, y} such that [x,k y] = 1, for some positive integer k. In this paper, we study finite groups G satisfying this condition. In particular, if G is a finitely generated soluble group, then G/Z*(G) ≤n113√n+2, where Z*(G) is the hypercentre of G. © 2004 Elsevier Inc. All rights reserved.
Publication Date: 2005
Communications in Algebra (00927872)33(9)pp. 3225-3238
A cover for a group G is a collection of proper subgroups whose union is the whole group G. A cover is irredundant if no proper sub-collection is also a cover, and is called maximal if all its members are maximal subgroups. For an integer n > 2, a cover with n members is called an n-cover. Also, we denote σ(G) = n if G has an n-cover and does not have any m-cover for each integer m > n. In this article, we completely characterize groups with a maximal irredundant 6-cover with core-free intersection. As an application of this result, we characterize the groups G with σ(G) = 6. The intersection of an irredundant n-cover is known to have index bounded by a function of n, though in general the precise bound is not known. We also prove that the exact bound is 36 when n is 6.
Publication Date: 2005
Algebra Colloquium (02191733)12(4)pp. 709-714
If K is a set of automorphisms of a group G, an endomorphism θ : G → G is said to be K-pointwise if for each element t ∈ G, there exists an element φ ∈ K such that θ(t) = φ(t). This generalizes the notion of pointwise inner automorphism. We show that in some special cases, a K-pointwise endomorphism is necessarily an automorphism (it is not true in general). © 2005 AMSS CAS & SUZHOU UNIV.
Publication Date: 2005
Semigroup Forum (00371912)71(3)pp. 471-480
We introduce the concept of paradoxical decomposition for semigroups. We show that a semigroup S admits a left paradoxical decomposition if and only if S is not left amenable. Also Rosenbelatt and Willis introduced the configuration concept for groups and showed that amenability of groups is equivalent with existence of normalised solution for any system of configuration equations. We generalise it for semigroups. © Springer 2005.
Publication Date: 2005
Journal of Pure and Applied Algebra (00224049)198(1-3)pp. 9-19
Let R be a ring and f(x1,..., xn) be a polynomial in noncommutative indeterminates x1,..., xn with coefficients from ℤ and zero constant. The ring R is said to be an f-ring if f(r1,..., rn) = 0 for all r1,..., rn of R and a virtually f-ring if for every n infinite subsets X1,..., Xn (not necessarily distinct) of R, there exist n elements r1 ∈ X1,...,rn ∈ Xn such that f(r1,..., rn) = 0. Let R* be the 'smallest' ring (in some sense) with identity containing R such that Char (R) = Char (R*). Then denote by ZR the subring generated by the identity of R* and denote by f̄R the image of f in ZR [x1,..., xn] (the ring of polynomials with coefficients in ZR in commutative indeterminates x1,..., xn). In this paper, we show that if R is a left primitive virtually f-ring such that f̄R ≠ 0, then R is finite. Using this result, we prove that an infinite semisimple virtually f-ring R is an f-ring, if the subring of ZR generated by the coefficients of f̄R is equal to ZR; and we also prove that if f(x) = ∑i=2n aixi + εx ∈ ℤ [x], where ε ∈ {-1, 1}, then every infinite virtually f-ring with identity is a commutative f-ring. Finally we show that a commutative Noetherian virtually f-ring R with identity is finite if the subring generated by the coefficients of f̄R is ZR. © 2004 Elsevier B.V. All rights reserved.
Publication Date: 2004
Rendiconti del Seminario Matematico dell 'Universita' di Padova/Mathematical Journal of the University of Padova (22402926)112pp. 173-180
Let n be an integer greater than 1. A group G is said to be n-rewritable (or a Qn-group) if for every n elements x1, x2,., xn in G there exist distinct permutations s and τ in Sn such that xσ(1)xσ(2)⋯xσ(n)=xτ(1)xτ(2)⋯xτ(n). In this paper we have completely characterized abelian-by-cyclic 3-rewritable groups: they turns out to have an abelian subgroup of index 2 or the size of derived subgroups is less than 6. In this paper, we also prove that G/F(G) is an abelian group of finite exponent dividing 12, where F(G) is the Fitting subgroup of G. © 2004, Universita di Padova. All rights reserved.
Publication Date: 2004
Illinois Journal of Mathematics (00192082)48(3)pp. 861-873
J. M. Rosenblatt and G. A. Willis introduced the notion of configurations for finitely generated groups G. They characterised amenability of G in terms of the configuration equations. In this paper we investigate which group properties can be characterised by configurations. It is proved that if G 1 and G2 are two finitely generated groups having the same configuration sets and G1 satisfies a semigroup law, then G 2 satisfies the same semigroup law. Furthermore, if G1 is abelian then G1 and G2 are isomorphic. ©2004 University of Illinois.
Publication Date: 2004
Journal of Pure and Applied Algebra (00224049)188(1-3)pp. 1-6
In this paper we study left 3-Engel elements in groups. In particular, we prove that for any prime p and any left 3-Engel element x of finite p-power order in a group G, xp is in the Baer radical of G. Also it is proved that 〈x,y〉 is nilpotent of class at most 4 for every two left 3-Engel elements in a group G. © 2003 Elsevier B.V. All rights reserved.
Publication Date: 2004
International Journal Of Mathematics And Mathematical Sciences (16870425)2004(7)pp. 373-375
We prove that for any odd integer N and any integer n>0, the Nth power of a product of n commutators in a nonabelian free group of countable infiniterank can be expressed as a product of squares of 2n+1 elements and, for all such odd N and integersn, there are commutators for which the number 2n+1 of squares is the minimum number such that the Nth power of its product can be written as a product of squares. This generalizes a recent result of Akhavan-Malayeri. Copyright © 2004 Hindawi Publishing Corporation. All rights reserved.
Publication Date: 2003
Bulletin of the Australian Mathematical Society (00049727)67(1)pp. 115-119
Let c ≥ 0, d ≥ 2 be integers and Nc(d) be the variety of groups in which every d-generator subgroup is nilpotent of class at most c. N.D. Gupta asked for what values of c and d is it true that Nc(d) is locally nilpotent? We prove that if c ≤ 2d + 2d-1 - 3 then the variety Nc(d) is locally nilpotent and we reduce the question of Gupta about the periodic groups in Nc(d) to the prime power exponent groups in this variety.
Publication Date: 2002
Proceedings of the American Mathematical Society (10886826)130(10)pp. 2827-2836
For a given positive integer n and a given prime number p, let r = r (n,p) be the integer satisfying pr-1 < n ≤ pr. We show that every locally finite p-group, satisfying the n-Engel identity, is (nilpotent of n-bounded class)-by-(finite exponent) where the best upper bound for the exponent is either pr or pr-1 if p is odd. When p = 2 the best upper bound is pr-1 pr or pr+1. In the second part of the paper we focus our attention on 4-Engel groups. With the aid of the results of the first part we show that every 4-Engel 3-group is soluble and the derived length is bounded by some constant.
Publication Date: 2002
Bulletin of the Belgian Mathematical Society - Simon Stevin (13701444)9(2)pp. 205-215
In this paper, we consider some combinatorial conditions on infinite subsets of groups, and we obtain in terms of these conditions some characterizations of nilpotent-by-finite and finite-by-nilpotent groups on the class of finitely generated soluble groups.
Publication Date: 2001
Bulletin of the Australian Mathematical Society (00049727)64(1)pp. 27-31
We use Ramsey's theorem to generalise a result of L. Babai and T.S. Sós on Sidon subsets and then use this to prove that for an integer n > 1 the class of groups in which every infinite subset contains a rewritable n-subset coincides with the class of groups in which every infinite subset contains n mutually disjoint non-empty subsets X1, ..., Xn such that X1 ⋯ Xn ∩ Xσ(1) ⋯ Xσ(n) ≠ 0 for some non-identity permutation σ on the set {1, ..., n}.
Publication Date: 2001
Communications in Algebra (00927872)29(4)pp. 1571-1581
Let n > 1 be an integer. A group G is said to be n-rewritable, whenever for any subset {x1, . . ., xn} of elements of G, there exist distinct permutations τ, σ of the set {1, 2, . . ., n} such that xτ(1) · · · xτ(n) = xσ(1) · · · xσ(n). In this paper we show that an infinite group G is n-rewritable if and only if for every n infinite subsets X1, . . ., Xn of G there exist distinct permutations τ, σ of the set {1, 2, . . ., n} such that Xτ(1) · · · Xτ(n) ∩ Xσ(n) · · · Xσ(n) ≠ 0.
Publication Date: 2001
Algebra Colloquium (02191733)8(2)pp. 153-157
Let α1,... ,αn ∈ ℕ. We prove that, in every infinite ring R, x1α1 ⋯xnαn = 0 for all x1,..., xn ∈ R if and only if, for any n infinite subsets X1,..., Xn of R, there exist x1 ∈ X1, ..., xn ∈ Xn such that x1α1 ⋯ xnαn = 0. © Inst. Math. CAS 2001.
Publication Date: 2001
Houston Journal of Mathematics (03621588)27(3)pp. 511-522
Let n and k be positive integers. We say that a group G satisfies the condition ε(n) (respectively, εk(n)) if and only if any set with n + 1 elements of G contains two distinct elements x, y such that [x,t y] = 1 for some positive integer t = t(x,y) (respectively, [x,k y] = 1). Here we study certain groups satisfying these conditions. We prove that if G is a finite group satisfying the condition ε(n), then G is nilpotent if n < 3 and G is soluble if n < 16. If G is a finitely generated soluble group satisfying the condition ε(2), then G is nilpotent. If k and n are positive integers and G is a finitely generated residually finite group satisfying the condition εk(n), then G is nilpotent if n < 3 and G is polycyclic if n < 16. In particular, there is a positive integer c depending only on k such that G/Zc(G) is finite, where Zc(G) is the (c + 1)-th term of the upper central series of G. Also these bounds cannot be improved.
Publication Date: 2000
Rendiconti del Seminario Matematico dell 'Universita' di Padova/Mathematical Journal of the University of Padova (22402926)104pp. 129-134
Let α1,…, αn be nonzero integers whose greatest common divisor is d. We prove that an infinite group G is of finite exponent dividing d if and only if for every n infinite subsets X1, …, Xn of G there exist x1 ∈ X1, …, xn ∈ Xn such that x1α1…xαnn = 1. © Rendiconti del Seminario Matematico della Università di Padova, 2000, tous droits réservés.
Publication Date: 2000
Rendiconti del Seminario Matematico dell 'Universita' di Padova/Mathematical Journal of the University of Padova (22402926)103pp. 47-49
In this note, we prove that, in every finitely generated soluble group G, G/Z2 (G) is finite if and only if in every infinite subset X of G there exist different x, y such that [x, y, y] = 1. © Rendiconti del Seminario Matematico della Università di Padova, 2000, tous droits réservés.
Publication Date: 2000
Bulletin of the Australian Mathematical Society (00049727)62(1)pp. 141-148
Let k be a positive integer. We denote by εk(∞) the class of all groups in which every infinite subset contains two distinct elements cursive Greek chi, y such that [cursive Greek chi,k y] = 1. We say that a group G is an ε*k-group provided that whenever X,Y are infinite subsets of G, there exists cursive Greek chi ∈ X, y ∈ Y such that [cursive Greek chi,k y] = 1. Here we prove that: (1) If G is a finitely generated soluble group, then G ∈ ε3(∞) if and only if G is finite by a nilpotent group in which every two generator subgroup is nilpotent of class at most 3. (2) If G is a finitely generated metabelian group, then G ∈ εk(∞) if and only if G/Zk(G) is finite, where Zk(G) is the (k + 1)-th term of the upper central series of G. (3) If G is a finitely generated soluble εk(∞)-group, then there exists a positive integer t depending only on k such that G/Zt(G) is finite. (4) If G is an infinite ε*k-group in which every non-trivial finitely generated subgroup has a non-trivial finite quotient, then G is k-Engel. In particular, G is locally nilpotent.
Publication Date: 1999
Archiv der Mathematik (0003889X)73(2)pp. 104-108
In this note we prove that every infinite group G is 3-abelian (i.e. (ab)3 = a3b3 for all a, b in G) if and only if in every two infinite subsets X and Y of G there exist x ∈ X and y ∈ Y such that (xy)3 = x3y3.
Publication Date: 1999
Communications in Algebra (00927872)27(11)pp. 5633-5638
In this note we show that if G is a finitely generated soluble group, then every infinite subset of G contains two elements generating a nilpotent group of class at most k if and only if G is finite by a group in which every two generator subgroup is nilpotent of class at most k.
Publication Date: 1999
Journal of Algebra (00218693)221(2)pp. 570-578
Let n be an integer greater than 1. A group G is said to be n-permutable whenever for every n-tuple (x1,...,xn) of elements of G there exists a non-identity permutation σ of {1,...,n} such that x1···xn=xσ(1)···xσ(n). In this paper we prove that an infinite group G is n-permutable if and only if for every n infinite subsets X1,...,Xn of G there exists a non-identity permutation σ on {1,...,n} such that X1···Xn∪Xσ(1)···Xσ(n)≠∅. © 1999 Academic Press.
