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

Let p be a prime and F-p be a finite field of p elements. Let F(p)G denote the group algebra of the finite p-group G over the field F-p and V (F(p)G) denote the group of normalized units in F(p)G. Suppose that G and H are finite p-groups given by a central extension of the form 1 -> Z(pm) -> G -> Z(p) x ... x Z(p) -> 1 and G' congruent to Z(p), m >= 1. Then V (F(p)G) congruent to V (FpH) if and only if G congruent to H. Balogh and Bovdi only solved the isomorphism problem when p is odd. In this paper, the case p = 2 is determined.