Quotients of Passman Fours Group and Non-units of Their Group Algebras

Abstract

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.