ON $\mathcal A_1$- AND $\mathcal A_2$-SUBGROUPS OF FINITE p-GROUPS

Abstract

The isomorphism types of all minimal nonabelian subgroups ([Formula: see text]-subgroups) of [Formula: see text]-groups are described. This allows us to classify the nonabelian p-groups, p > 2, that have no p isomorphic [Formula: see text]-subgroups of minimal order. In particular, if a p-group G is neither abelian nor [Formula: see text]-group and all its [Formula: see text]-subgroups are pairwise non-isomorphic, then either G ≅ SD 16, the semidihedral group of order 16, or G is a metacyclic 2-group of order ≥ 25. 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 [Formula: see text]-subgroups are metacyclic.