The unitary subgroups of group algebras for a class of finite 2-groups with the derived subgroup of order 2

Let p be a prime and F be a finite field of characteristic p. Suppose that FG is the group algebra of the finite p-group G over the field F. Let V (FG) denote the group of normalized units in FG and let V_*(FG) denote the unitary subgroup of V (FG). If p is odd, then the order of V_*(FG) is ∣F∣^{(∣G∣−1)/2}. However, the case p = 2 still is open. In this paper, the order of V_*(FG) is computed when G is a nonabelian 2-group given by a central extension of the form

\(1 \to {\mathbb{Z}_{{2^n}}} \times {\mathbb{Z}_{{2^m}}} \to G \to {\mathbb{Z}_2} \times \cdots \times {\mathbb{Z}_2} \to 1\) $$1\to {\mathbb{Z}}_{2^n}\times {\mathbb{Z}}_{2^m}\to G\to {\mathbb{Z}}_2\times \cdots \times {\mathbb{Z}}_2\to 1$$

and G′ ≅ ℤ₂, n, m ⩾ 1. Furthermore, a conjecture is confirmed, i.e., the order of V_*(FG) can be divisible by \(|F{|^{{1 \over 2}(|G| + |{\Omega _1}(G)|) - 1}}\) $$\mid F{\mid }^{\frac{1}{2}(\mid G\mid +\mid {\Omega }_1(G)\mid )-1}$$ , where Ω₁(G) = {g ∈ G ∣ g² = 1}.