On finite p-groups with few nonabelian subgroups of order p ^p and exponent p

Abstract

Let G be a finite p-group. If p = 2, then a nonabelian group G = Ω₁(G) is generated by dihedral subgroups of order 8. If p > 2 and a nonabelian group G = Ω₁(G) has no subgroup isomorphic to ${\Sigma _{{p^2}}}$, a Sylow p-subgroup of the symmetric group of degree p ², then it is generated by nonabelian subgroups of order p ³ and exponent p. If p > 2 and the irregular p-group G has < p nonabelian subgroups of order p ^p and exponent p, then G is of maximal class and order p ^p+1. We also study in some detail the p-groups, containing exactly p nonabelian subgroups of order p ^p and exponent p. In conclusion, we prove three new counting theorems on the number of subgroups of maximal class of certain type in a p-group. In particular, we prove that if p > 2, and G is a p-group of order > p ^p+1, then the number of subgroups ≅ Σ${\Sigma _{{p^2}}}$ in G is a multiple of p.