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

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.