The order of the unitary subgroups of group algebras

Abstract

Let F G be the group algebra of a finite p -group G over a finite field F of positive characteristic p . Let ⊛ be an involution of the algebra F G which is a linear extension of an anti-automorphism of the group G to F G . If p is an odd prime, then the order of the ⊛ -unitary subgroup of F G 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 F G 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 F G determines the order of the finite p -group G .