The order of the unitary subgroups of group algebras

Abstract

Let [Formula: see text] be the group algebra of a finite [Formula: see text]-group [Formula: see text] over a finite field [Formula: see text] of positive characteristic [Formula: see text]. Let ⊛ be an involution of the algebra [Formula: see text] which is a linear extension of an anti-automorphism of the group [Formula: see text] to [Formula: see text]. If [Formula: see text] is an odd prime, then the order of the ⊛-unitary subgroup of [Formula: see text] is established. For the case [Formula: see text] we generalize a result obtained for finite abelian [Formula: see text]-groups. It is proved that the order of the ∗-unitary subgroup of [Formula: see text] of a non-abelian [Formula: see text]-group is always divisible by a number which depends only on the size of [Formula: see text], the order of [Formula: see text] and the number of elements of order two in [Formula: see text]. Moreover, we show that the order of the ∗-unitary subgroup of [Formula: see text] determines the order of the finite [Formula: see text]-group [Formula: see text].