Symmetric subgroups in modular group algebras

TL;DR

This paper investigates symmetric subgroups within the normalized unit group of modular group algebras of finite p-groups, introduces their properties, and provides a counterexample to a longstanding conjecture.

Contribution

It defines symmetric subgroups in V(KG), analyzes their properties, and disproves Bovdi's conjecture regarding their generation of the unit group.

Findings

Symmetric subgroups are invariant under the classical involution.

Counterexample disproves Bovdi's conjecture that V(KG) is generated by G and symmetric units.

The study advances understanding of the structure of modular group algebra units.

Abstract

Let V(KG) be a normalised unit group of the modular group algebra of a finite p-group G over the field K of p elements. We introduce a notion of symmetric subgroups in V(KG) as subgroups invariant under the action of the classical involution of the group algebra KG. We study properties of symmetric subgroups and construct a counterexample to the conjecture by V.Bovdi, which states that V(KG)=<G,S*>, where S* is a set of symmetric units of V(KG).