Abstract. We determine the number of elements of order two in the group of normalized units V (F 2 G) of the group algebra F 2 G of a 2-group of maximal class over the field F 2 of two elements. As a consequence for the 2-groups G and H of maximal class we have that V (F 2 G) and V (F 2 H) are isomorphic if and only if G and H are isomorphic. Notation and results.Let G be a finite p-group and F p G its group algebra over the fieldis called the group of normalized units. Evidently V (F p G) is a p-group and its order is p |G|−1 .Let C = a | a 2 n = 1 be the cyclic group of order 2 n , where n ≥ 2. Consider the following extensions of C:which are the generalized quaternion group, the dihedral group and the semidihedral group respectively. It is well-known that a finite 2-group of maximal class coincides with one of these groups.

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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.