Abdollahi, A.,
Bagherian, J.,
Jafari, F.,
Khatami bidgoli, M.,
Parvaresh, F.,
Sobhani, R. 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.
Algebras and Representation Theory (15729079)26(6)pp. 3085-3100
Let G be a finite group and χ be an irreducible character of G, the number cod (χ) = | G: ker(χ) | / χ(1 ) is called the codegree of χ. Also, cod(G) = {cod(χ) | χ ∈Irr(G)}. For d ∈cod(G), the multiplicity of d in G, denoted by mG′(d), is the number of irreducible characters of G having codegree d. A finite group G is called a Tk′-group for some integer k ≥ 1, if there exists d0 ∈cod(G) such that mG′(d0)=k and for every d ∈cod(G) −{d0}, we have mG′(d)=1. In this note we characterize finite Tk′-groups completely, where k ≥ 1 is an integer. © 2022, The Author(s), under exclusive licence to Springer Nature B.V.
DISCRETE MATHEMATICS (0012365X)346(1)
Let G be a finite group and Irr(G) be the set of all complex irreducible characters of G. The character-graph Delta(G) associated to G, is a graph whose vertex set is the set of primes which divide the degrees of some characters in Irr(G) and two distinct primes p and q are adjacent in Delta(G) if the product pq divides x(1), for some x is an element of Irr(G). Tong-Viet posed the conjecture that if Delta(G) is k-regular for some integer k ? 2, then Delta(G) is either a complete graph or a cocktail party graph. In this paper, we show that his conjecture is true for all regular character-graphs whose eigenvalues are in the interval [-2, infinity).(c) 2022 Elsevier B.V. All rights reserved.
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.
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.
Abdollahi, A.,
Bagherian, J.,
Ebrahimi m., M.,
Garmsiri f.m., F.M.,
Khatami bidgoli, M.,
Sobhani, R. 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.
Abdollahi, A.,
Bagherian, J.,
Ebrahimi m., M.,
Khatami bidgoli, M.,
Shahbazi, Z.,
Sobhani, R. 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.
Journal of Algebra and its Applications (17936829)20(4)
For a finite group G, let Δ(G) be the character-graph which is built on the set of irreducible complex character degrees of G. In this paper, we wish to determine the structure of finite groups G such that Δ(G) is 1-connected with nonbipartite complement. Also, we classify all 1-connected graphs with nonbipartite complement that can occur as the character-graph Δ(G) of a finite group G. © 2021 World Scientific Publishing Company.
Communications in Algebra (00927872)49(9)pp. 4079-4087
For a finite group G, let R(G) be the solvable radical of G. The character-graph (Formula presented.) of G is a graph whose vertices are the primes which divide the degrees of some irreducible complex characters of G and two distinct primes p and q are joined by an edge if the product pq divides some character degree of G. In this paper we prove that, if (Formula presented.) has no subgraph isomorphic to (Formula presented.) and it’s complement is non-bipartite, then (Formula presented.) is an almost simple group with socle isomorphic to (Formula presented.) where (Formula presented.) is a prime power. Also we study the structure of all planar graphs that occur as the character-graph (Formula presented.) of a finite group G. © 2021 Taylor & Francis Group, LLC.
Mathematical Reports (15823067)21(4)pp. 431-440
Let G be a finite group, and Irr(G) be the set of complex irreducible characters of G. An element g ∈ G is called a vanishing element if there exists an irreducible character χ ∈ Irr(G) such that χ(g) = 0. The set of orders of vanishing elements of G is denoted by Vo(G). A recent conjecture states that if G is a finite group and M is a finite nonabelian simple group such that Vo(G) = Vo(M) and (G) = (M), then G ≅ M. In this paper, we give a positive answer to this conjecture for a family of classical simple groups, namely Ap(2) and Ap-1(2), where p ≠ 2; 3 and 2p-1 is a prime. © 2019 Editura Academiei Romane. All rights reserved.
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.
Quasigroups and Related Systems (15612848)27(1)pp. 15-24
Let G be a group, and πe(G) be the set of element orders of G. For k ϵ πe(G), the number of elements of G of order k is denoted by mk (G). Set nse(G) = {mk (G) | k ϵ πe(G)}. Let q = 22n+1, and p = q-1 be a Mersenne prime. In this paper, we show that if G is a group such that nse(G) = nse(Sz(q)) and p ϵ πe(G) but p2 ∉ πe(G), then G≅= Sz(q) or G≅= Sz(q) ⋊ ℤ2n+1. © 2019, Institute of Mathematics, Academy of Sciences Moldova. All rights reserved.
International Journal Of Group Theory (22517669)8(2)pp. 41-46
Abstract. Let G be a finite group, and Irr(G) be the set of complex irreducible characters of G. Let A.(G) be the set of prime divisors of character degrees of G. The character degree graph of G, which is denoted by δ(G), is a simple graph with vertex set A.(G), and we join two vertices r and s by an edge if there exists a character degree of G divisible by rs. In this paper, we prove that if G is a finite group such that δ(G) = δ(PSL2(q)) and |G| = |PSL2(q)|, then G = PSL2(q). © 2019 University of Isfahan.
Czechoslovak Mathematical Journal (00114642)68(1)pp. 121-130
Let G be a finite group. An element g ∈ G is called a vanishing element if there exists an irreducible complex character χ of G such that χ(g)= 0. Denote by Vo(G) the set of orders of vanishing elements of G. Ghasemabadi, Iranmanesh, Mavadatpour (2015), in their paper presented the following conjecture: Let G be a finite group and M a finite nonabelian simple group such that Vo(G) = Vo(M) and |G| = |M|. Then G ≌ M. We answer in affirmative this conjecture for M = Sz(q), where q = 22n+1 and either q − 1, q−2q+1 or q + 2q+1 is a prime number, and M = F4(q), where q = 2n and either q4 + 1 or q4 − q2 + 1 is a prime number. © 2018, Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic.
Communications in Algebra (00927872)44(9)pp. 3927-3932
Let G be a finite group and cs(G) be the set of conjugacy class sizes of G. In 1987, J. G. Thompson conjectured that, if G is a finite group with Z(G) = 1 and M is a nonabelian simple group satisfying that cs(G) = cs(M), then G ≅ M. This conjecture has been proved for Suzuki groups in [5]. In this article, we improve this result by proving that, if G is a finite group such that cs(G) = cs(Sz(q)), for q = 22m+1, then G ≅ Sz(q) × A, where A is abelian. We avoid using classification of finite simple groups in our proofs. © 2016, Copyright © Taylor & Francis Group, LLC.
Journal of Group Theory (14354446)18(1)pp. 115-131
In [1], a conjecture of J. G. Thompson for PSLn(q) was proved. It was shown that every finite group G with the property Z(G) = 1 and cs(G) = cs(PSLn(q)) is isomorphic to PSLn(q) where cs(G) is the set of conjugacy class sizes of G . In this article we improve this result for PSL2(q). In fact we prove that if cs(G) = cs(PSL2(q)), for q > 3, then G ≅ PSL2(q) x A, where A is abelian. Our proof does not depend on the classification of finite simple groups. © de Gruyter 2015.
Proceedings of the Edinburgh Mathematical Society (14643839)56(2)pp. 371-386
Let G be a finite p-solvable group. We describe the structure of the p-complements of G when the set of p-regular conjugacy classes has exactly three class sizes. For instance, when the set of p-regular class sizes of G is {1, pa , pam} or {1, m, pam} with (m, p) = 1, then we show that m = qb for some prime q and the structure of the p-complements of G is determined. Copyright © Edinburgh Mathematical Society 2013.
Quaestiones Mathematicae (1727933X)36(4)pp. 517-535
Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, p′ are joined by an edge if G has an element of order pp′. Let L = Dn(q) have disconnected prime graph and q {small element of} {2, 5}. In this paper, we determine finite groups G with the same prime graph as L, i.e. Γ(G) = Γ(L). © 2013 © 2013 NISC (Pty) Ltd.
Monatshefte fur Mathematik (14365081)167(1)pp. 1-12
Let G be a finite group and N be a normal subgroup of G. Suppose that the set of G-conjugacy class sizes of N is {1, m, n}, with m < n and m does not divide n. In this paper, we show that N is solvable, and we determine the structure of these subgroups. © 2011 Springer-Verlag.
Monatshefte fur Mathematik (14365081)163(1)pp. 39-50
Let G be a group and πe(G) be the set of element orders of G. Let ε πe(G) and mk be the number of elements of order k in G. Let nse(G) = {mk εe(G). In Shen et al. (Monatsh Math, 2009), the authors proved that A4 ≅ PSL(2,3), A5 ≅ PSL(2,4) ≅ PSL(2,5) and A6 ≅ PSL(2,9) are uniquely determined by nse(G). In this paper, we prove that if G is a group such that nse(G) = nse(PSL(2, q)), where q ε {7,8,11,13}, then G ≅ PSL(2,q). © 2009 Springer-Verlag.
Bulletin Of The Malaysian Mathematical Sciences Society (01266705)34(3)pp. 665-674
Let G be a finite non-abelian group. The noncommuting graph of G is denoted by ∇(G) and is defined as follows: the vertex set of ∇ (G) is G\Z(G) and two vertices x and y are adjacent if and only if xy ≠ yx. Let p be a prime number. In this paper, it is proved that the almost simple group PGL(2, p) is uniquely determined by its noncommuting graph. As a consequence of our results the validity of a conjecture of Thompson and another conjecture of Shi and Bi for the group PGL(2,p) are proved.
Rocky Mountain Journal of Mathematics (00357596)41(5)pp. 1523-1545
Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, p' are joined by an edge if there is an element in G of order pp'. Let p be a prime number. In [4], the authors determined the structure of finite groups with the same element orders as PGL(2,p), and it is proved that there are infinitely many nonisomorphic finite groups with the same element orders as PGL(2,p). Therefore there are infinitely many nonisomorphic finite groups with the same prime graph as PGL{2,p). We know that PGL(2,p) has a unique nonabelian composition factor which is isomorphic to PSL(2,p). Let p be a prime number which is not a Mersenne or Fermat prime and p ≠11, 19. In this paper we determine the structure of finite groups with the same prime graph as PGL(2,p) and as the main result we prove that if G is a finite group such that Γ(G) = Γ(PGL(2,p)) and p≠ 13, then G has a unique nonabelian composition factor which is isomorphic to PSL(2,p) and if p = 13, then G has a unique nonabelian composition factor which is isomorphic to PSL(2,13) or PSL(2,27). Copyright © 2011 Rocky Mountain Mathematics Consortium.
Journal of Algebra (00218693)336(1)pp. 236-241
Let G be a finite p-solvable group and N be a normal subgroup of G. Suppose that the p-regular elements of N have exactly two G-conjugacy class sizes. In this paper it is shown that, if H is a p-complement of N, then either H is abelian or H is a product of a q-group for some prime q{double barred pipep and a central subgroup of G. © 2011 Elsevier Inc.
Publicationes Mathematicae Debrecen (00333883)78(2)pp. 469-484
Let G be a finite group. The prime graph (G) of G is defined as follows. The vertices of (G) are the primes dividing the order of G and two distinct vertices p and p′ are joined by an edge if there is an element in G of order pp′. It is proved that Dn(q), with disconnected prime graph, is quasirecognizable by their element orders. In this paper as the main result, we show that Dn(3), where n € {p,p+ 1} for an odd prime p > 3, is quasirecognizable by its prime graph.
International Journal of Algebra and Computation (02181967)20(7)pp. 847-873
Let G be a finite group. The prime graph Γ(G) of G is defined as follows. The vertices of Γ(G) are the primes dividing the order of G and two distinct vertices p, p′ are joined by an edge if there is an element in G of order pp′. In [G. Y. Chen et al., Recognition of the finite almost simple groups PGL2(q) by their spectrum, Journal of Group Theory, 10 (2007) 7185], it is proved that PGL(2, pk), where p is an odd prime and k > 1 is an integer, is recognizable by its spectrum. It is proved that if p > 19 is a prime number which is not a Mersenne or Fermat prime and Γ(G) = Γ(PGL(2, p)), then G has a unique nonabelian composition factor which is isomorphic to PSL(2, p). In this paper as the main result, we show that if p is an odd prime and k > 1 is an odd integer, then PGL(2, pk) is uniquely determined by its prime graph and so these groups are characterizable by their prime graphs. © 2010 World Scientific Publishing Company.
Acta Mathematica Hungarica (15882632)122(4)pp. 387-397
As the main result, we show that if G is a finite group such that Γ(G) = Γ(2 F 4(q)), where q = 22m+1 for some m 1, then G has a unique nonabelian composition factor isomorphic to 2 F 4(q). We also show that if G is a finite group satisfying |G| =|2 F 4(q)| and Γ(G) = Γ( 2 F 4(q)), then G 2 F 4(q). As a consequence of our result we give a new proof for a conjecture of W. Shi and J. Bi for 2 F 4(q). © 2009 Springer Science+Business Media B.V.
