The unitary subgroups of group algebras of a class of finite p-groups

Abstract

Let p be a prime and let F be a finite field of characteristic p. Let FG denote the group algebra of the finite p-group G over the field F and let [Formula: see text] denote the group of normalized units in FG. The anti-automorphism [Formula: see text] of G extends linearly to an anti-automorphism [Formula: see text] of FG. An element [Formula: see text] is called unitary if [Formula: see text]. All unitary elements of [Formula: see text] form a subgroup which is denoted by [Formula: see text]. If p is odd, the order of [Formula: see text] is [Formula: see text]. However, to compute the order of [Formula: see text] still is open when [Formula: see text]. In this paper, the order of [Formula: see text] is computed when G is a nonabelian [Formula: see text]-group given by a central extension of the following form: [Formula: see text] and [Formula: see text], [Formula: see text]. Further, a conjecture is confirmed, namely, the order of [Formula: see text] can be divided by [Formula: see text], where [Formula: see text].