The Order of 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 positive characteristic $p$. Let $\cd$ be an involution of the algebra $FG$ which is a linear extension of an anti-automorphism of the group $G$ to $FG$. If $p$ is an odd prime, then the order of the $\cd$-unitary subgroup of $FG$ is established. For the case $p=2$ we generalize a result obtained for finite abelian $2$-groups. It is proved that the order of the $*$-unitary subgroup of $FG$ of a non-abelian $2$-group is always divisible by a number which depends only on the size of $F$, the order of $G$ and the number of elements of order two in $G$. Moreover, we show that the order of the $*$-unitary subgroup of $FG$ determines the order of the finite $p$-group $G$.