A class of finite p-groups and the normalized unit groups of group algebras

Abstract

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 is a finite p-group given by a central extension of the form 1 → Z p n × Z p m → G → Z p × ⋯ × Z p → 1 and G ′ ≅ Z p , n , m ≥ 1 and p is odd. In this paper, the structure of G is determined. And the relations of V ( F p G ) p l and G p l , Ω l ( V ( F p G ) ) and Ω l ( G ) are given. Furthermore, there is a direct proof for V ( F p G ) p ∩ G = G p .