The Order of the Unitary Subgroups of Group Algebras

TL;DR

This paper investigates the size of unitary subgroups in group algebras over finite fields, providing formulas and divisibility results that connect subgroup order to properties of the underlying group.

Contribution

It establishes explicit formulas for the order of unitary subgroups in group algebras of finite p-groups, extending known results to non-abelian 2-groups and linking subgroup order to group structure.

Findings

Order of the $ ext{ extcyrillic}$-unitary subgroup is determined for odd primes.

Divisibility properties of the unitary subgroup order for 2-groups are established.

The order of the unitary subgroup uniquely determines the order of the underlying p-group.

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 $G$.