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

Abstract

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 $V_{\ast }(FG)$ denote 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 when $p=2$ still is open. In this paper, the order of $V_{\ast }(FG)$ is computed when $G$ is a nonabelian $2$-group given by a central extension of the form $$1⟶{\mathbb{Z}}_{2^n}\times {\mathbb{Z}}_{2^m}⟶G⟶{\mathbb{Z}}_2\times \cdots \times {\mathbb{Z}}_2⟶1$$ and $G^{\prime }\cong {\mathbb{Z}}_2$, $n,m\ge 1$. Further, a conjecture is confirmed, namely, the order of $V_{\ast }(FG)$ can be divisible by $|F{|}^{\frac{1}{2}(|G|+|{\Omega }_1(G)|)-1}$, where ${\Omega }_1(G)=\{g\in G|g^2=1\}$.

29pages. It has been accepted for publication in SCIENCE CHINA Mathematics