Nonnormal and minimal nonabelian subgroups of a finite group |
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Authors: | Yakov Berkovich |
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Institution: | 1.Department of Mathematics,University of Haifa Mount Carmel,Haifa,Israel |
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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 Sp2{\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 ≅ ΣSp2{\Sigma _{{p^2}}} in G is a multiple of p. |
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Keywords: | |
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