The Isomorphism Problem of Normalized Unit Groups of Group Algebras of a Class of Finite 2-groups

Abstract

Let p be a prime and ${\mathbb{F}_p}$ be a finite field of p elements. Let ${\mathbb{F}_p}G$ denote the group algebra of the finite p-group G over the field ${\mathbb{F}_p}$ and $V({\mathbb{F}_p}G)$ denote the group of normalized units in ${\mathbb{F}_p}G$. Suppose that G and H are finite p-groups given by a central extension of the form

$$1 \to {\mathbb{Z}_{{p^m}}} \to G \to {\mathbb{Z}_p} \times \cdots \times {\mathbb{Z}_p} \to 1$$

and ${G^\prime } \cong {\mathbb{Z}_p},\,\,m \ge 1$. Then $V({\mathbb{F}_p}G) \cong V({\mathbb{F}_p}H)$ if and only if G ≅ H. Balogh and Bovdi only solved the isomorphism problem when p is odd. In this paper, the case p = 2 is determined.