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

Let p be a prime and Fp be a finite field of p elements. Let FpG denote the group algebra of the finite p-group G over the field Fp and V(FpG) denote the group of normalized units in FpG. Suppose that G is a finite p-group given by a central extension of the form 1→Zpn×Zpm→G→Zp×⋯×Zp→1 and G′≅Zp, n,m≥1 and p is odd. In this paper, the structure of G is determined. And the relations of V(FpG)pl and Gpl, Ωl(V(FpG)) and Ωl(G) are given. Furthermore, there is a direct proof for V(FpG)p∩G=Gp.