Gregory T. Lee

Group algebras whose symmetric and skew elements are Lie solvable

Abstract Let FG be the group algebra of a group G without 2-elements over a field F of characteristic p ≠ 2 endowed with the canonical involution induced from the map g ↦ g –1, g ∈ G. Let (FG)– and (FG)+ be the sets of skew and symmetric elements of FG, respectively, and let P denote the set of p-elements of G (with P = 1 if p = 0). In the present paper we prove that if either P is finite or G is non-torsion and (FG)– or (FG)+ is Lie solvable, then FG is Lie solvable. The remaining cases are also settled upon small restrictions.

Group rings whose symmetric units are nilpotent

Abstract Let K be an infinite field of characteristic different from 2, and G a group containing elements of infinite order. We classify the groups G such that the symmetric units of KG satisfy the identity (x 1, x 2, …, x n ) = 1, for some n.

Group Rings Whose Symmetric Units are Solvable

Let K be an infinite field of characteristic different from 2, and G a group. Under suitable restrictions upon G, we classify the groups such that the symmetric units of KG satisfy the solvability identity (x 1, x 2,…, x 2 n ) o = 1, for some n.

A Survey on *-Group Identities on Units of Group Rings

Let F be an infinite field of characteristic different from 2 and G a torsion group. Let the group ring FG have an involution induced from an involution on G. We survey recent results establishing the conditions under which the unit group of FG satisfies a *-group identity. The surprising outcome is that the unit group satisfies a *-group identity if and only if the symmetric units of FG satisfy a group identity.

Group Rings Whose Symmetric Units Generate an n-Engel Group

Let F be an infinite field of characteristic different from 2 and G a torsion group. Write 𝒰+(FG) for the set of units in the group ring FG that are symmetric with respect to the classical involution induced from the map g ↦ g −1, for all g ∈ G. We classify the groups such that ⟨𝒰+(FG)⟩ is n-Engel.

Torsion matrices over commutative integral group rings

Let

Public Key Cryptography

{\mathbb Z}A

Ideals, Factor Rings and Homomorphisms

be the integral group ring of a finite abelian group

Introduction to Rings

A

Introduction to Groups

, and