On the isomorphism of unitary subgroups of noncommutative group algebras

Abstract

Let $FG$ be the group algebra of a finite $p$-group $G$ over a field $F$ of characteristic $p$. Let $\circledast$ be an involution of the group algebra $FG$ which arises form the group basis $G$. The upper bound for the number of non-isomorphic $\circledast$-unitary subgroups is the number of conjugacy classes of the automorphism group $G$ with all the elements of order two. The upper bound is not always reached in the case when $G$ is an abelian group, but for non-abelian case the question is open. In this paper we present a non-abelian $p$-group $G$ whose group algebra $FG$ has sharply less number of non-isomorphic $\circledast$-unitary subgroups than the given upper bound.

Received: 18 December 2021 | Accepted: 8 January 2022