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 ≅ ��\({\Sigma _{{p^2}}}\) in G is a multiple of p.
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Berkovich, Y. On finite p-groups with few nonabelian subgroups of order p p and exponent p . Isr. J. Math. 179, 189–210 (2010). https://doi.org/10.1007/s11856-010-0078-x
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DOI: https://doi.org/10.1007/s11856-010-0078-x