Group algebras with unit group of class $p$

Abstract

Let V (F p G) be the group of normalized units of the group algebra F p G of a finite nonabelian p-group G over the field F p of p elements. Our goal is to investigate the power structure of V (F p G), when it has nilpotency class p. As a consequence, we have proved that if G and H are p-groups with cyclic Frattini subgroups and p > 2, then V (F p G) is isomorphic to V (F p H) if and only if G and H are isomorphic.