Group algebras with unit group of class p

Let V(script F sign _pG) be the group of normalized units of the group algebra script F sign _pG of a finite nonabelian p-group G over the field script F sign _p of p elements. Our goal is to investigate the power structure of V(script F sign _pG), 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(script F sign _pG) is isomorphic to V(script F sign _pH) if and only if G and H are isomorphic.