ON A1- AND A2-SUBGROUPS OF FINITE p-GROUPS

The isomorphism types of all minimal nonabelian subgroups (- A(1-)subgroups) of A(2)-groups are described. This allows us to classify the nonabelian p-groups, p > 2, that have no p isomorphic A(1)-subgroups of minimal order. In particular, if a p-group G is neither abelian nor A(1)-group and all its A(1)-subgroups are pairwise non-isomorphic, then either G congruent to SD16, the semidihedral group of order 16, or G is a metacyclic 2-group of order >= 2(5). We also show that if a p-group G is neither abelian nor minimal nonabelian, then G is metacyclic if and only if all its A(2)-subgroups are metacyclic.