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OVERGROUPS OF WEAK SECOND MAXIMAL SUBGROUPS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 99 / Issue 1 / February 2019
- Published online by Cambridge University Press:
- 30 August 2018, pp. 83-88
- Print publication:
- February 2019
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- Article
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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].
