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 p2, then it is generated by nonabelian subgroups of order p3 and exponent p. If p > 2 and the irregular p-group G has 2, and G is a p-group of order > pp+1, then the number of subgroups ≅ Σ\({\Sigma _{{p^2}}}\) in G is a multiple of p.