On the isomorphism of unitary subgroups of noncommutative group algebras

Let F G be the group algebra of a finite p-group G over a field F of characteristic p. Let ⊛ be an involution of the group algebra F G which arises form the group basis G. The upper bound for the number of non-isomorphic ⊛-unitary subgroups is the number of conjugacy classes of the automorphism group G with all the elements of order two. The upper bound is not always reached in the case when G is an abelian group, but for non-abelian case the question is open. In this paper we present a non-abelian p-group G whose group algebra F G has sharply less number of non-isomorphic ⊛-unitary subgroups than the given upper bound.