On the isomorphism of unitary subgroups of noncommutative group algebras

: Let FG be the group algebra of a finite p -group G over a field F of characteristic p . Let (cid:126) be an involution of the group algebra FG which arises form the group basis G . The upper bound for the number of non-isomorphic (cid:126) -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 FG has sharply less number of non-isomorphic (cid:126) -unitary subgroups than the given upper bound.