Symmetric units in modular group algebras

TL;DR

This paper investigates the conditions under which the symmetric units in modular group algebras form a multiplicative group, focusing on locally finite p-groups over rings of characteristic p.

Contribution

It characterizes the groups G and rings K for which the symmetric units in KG constitute a multiplicative group.

Findings

Identifies specific conditions on G and K for symmetric units to form a group.

Provides a classification of symmetric units in modular group algebras.

Enhances understanding of involution-invariant elements in group algebras.

Abstract

Let p be a prime, G a locally finite p-group, K a commutative ring of characteristic p. The anti-automorphism g\mapsto g\m1 of G extends linearly to an anti-automorphism a\mapsto a^* of KG. An element a of KG is called symmetric if a^*=a. In this paper we answer the question: for which G and K do the symmetric units of KG form a multiplicative group.