On the Unitary Subgroups of group algebras

Abstract

Let $FG$ be the group algebra of a finite $p$-group $G$ over a finite field $F$ of characteristic $p$ and $\ast$ the classical involution of $FG$. The $\ast$-unitary subgroup of $FG$, denoted by $V_{\ast }(FG)$, is defined to be the set of all normalized units $u$ satisfying the property $u^{\ast }=u^{-1}$. In this paper we give a recursive method how to compute the order of the $\ast$-unitary subgroup for many non-commutative group algebras. We also prove a variant of the modular isomorphism question of group algebras, where $F$ is a finite field of characteristic two, that is $V_{\ast }(FG)$ determines the basic group $G$ for all non-abelian $2$-groups $G$ of order at most $2^4$.