Group algebras whose involutory units commute

TL;DR

This paper characterizes nonabelian locally finite 2-groups over fields of characteristic 2 for which all involutory units in their group algebra commute, providing an explicit classification independent of the field.

Contribution

It provides a complete classification of groups G where involutions in V(KG) commute, regardless of the field K, for nonabelian locally finite 2-groups.

Findings

All involutions in V(KG) commute if and only if G is on a specific list of groups.

The property depends solely on G, not on the field K.

Explicit list of groups G with this property is provided.

Abstract

Let K be a field of characteristic 2 and G a nonabelian locally finite 2-group. Let V(KG) be the group of units with augmentation 1 in the group algebra KG. An explicit list of groups is given, and it is proved that all involutions in V(KG) commute with each other if and only if G is isomorphic to one of the groups on this list. In particular, this property depends only on G and not at all on K.