The unitary subgroups of group algebras for a class of finite \(2\)-groups with the derived subgroup of order \(2\) (Q6608126)

Let \(p\) be a prime, \(G\) a finite \(p\)-group and \(F\) a a finite field of characteristic \(p\). If \(FG\) is the group algebra \(G\) over \(F\), let \(V(FG)\) be the group of normalized units in \(FG\) and let \(V_{\ast}(FG)\) be the unitary subgroup of \(V(FG)\). If \(p\) is odd, then the order of \(V_{\ast}(FG)\) is \(|F|^{(|G|-1)/2}\), however, the case \(p=2\) is still open.\N\NIn the paper under review, the authors compute the order of \(V_{\ast}(FG)\) if \(G\) is a \(2\)-group given by a central extension of the form \(1 \rightarrow \mathbb{Z}_{2^{n}} \times \mathbb{Z}_{2^{n}} \rightarrow G \rightarrow \mathbb{Z}_{2} \times \dots \times \mathbb{Z}_{2} \rightarrow 1\) and such that \(G' \simeq \mathbb{Z}_{2}\). In particular, they prove that the order of \(V_{\ast}(FG)\) is divisible by \(|F|^{\frac{1}{2}(|G|+|\Omega_{1}(G)|)-1}\), confirming a conjecture proposed by \textit{A. A. Bovdi} and \textit{A. A. Sakach}, Math. Notes 45, No. 6, 445--450 (1989; Zbl 0688.16008).