ON FINITE p-GROUPS WITH FEW NONABELIAN SUBGROUPS OF ORDER pp AND EXPONENT p

Let G be a finite p-group. If p = 2, then a nonabelian group G = Omega(1)(G) is generated by dihedral subgroups of order 8. If p > 2 and a nonabelian group G = Omega(1)(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 similar or equal to Sigma(p2) in G is a multiple of p.