The order of the unitary subgroups of group algebras

Let FG 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 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 -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.