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OVERGROUPS OF WEAK SECOND MAXIMAL SUBGROUPS

Published online by Cambridge University Press:  30 August 2018

HANGYANG MENG*
Affiliation:
Department of Mathematics, Shanghai University, Shanghai 200444, PR China email [email protected]
XIUYUN GUO
Affiliation:
Department of Mathematics, Shanghai University, Shanghai 200444, PR China email [email protected]
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Abstract

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A subgroup $H$ is called a weak second maximal subgroup of $G$ if $H$ is a maximal subgroup of a maximal subgroup of $G$. Let $m(G,H)$ denote the number of maximal subgroups of $G$ containing $H$. We prove that $m(G,H)-1$ divides the index of some maximal subgroup of $G$ when $H$ is a weak second maximal subgroup of $G$. This partially answers a question of Flavell [‘Overgroups of second maximal subgroups’, Arch. Math.64(4) (1995), 277–282] and extends a result of Pálfy and Pudlák [‘Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups’, Algebra Universalis11(1) (1980), 22–27].

Type
Research Article
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

The research for this work was partially supported by the National Natural Science Foundation of China (11771271).

References

Flavell, P., ‘Overgroups of second maximal subgroups’, Arch. Math. 64(4) (1995), 277282.Google Scholar
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Meng, H. and Guo, X., ‘Weak second maximal subgroups in solvable groups’, arXiv:1808.02309.Google Scholar
Pálfy, P. P. and Pudlák, P., ‘Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups’, Algebra Universalis 11(1) (1980), 2227.Google Scholar