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GROUPS WHOSE NONNORMAL SUBGROUPS ARE METAHAMILTONIAN
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 102 / Issue 1 / August 2020
- Published online by Cambridge University Press:
- 08 October 2019, pp. 96-103
- Print publication:
- August 2020
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- Article
- Export citation
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If
$\mathfrak{X}$ is a class of groups, we define a sequence 
$\mathfrak{X}_{1},\mathfrak{X}_{2},\ldots ,\mathfrak{X}_{k},\ldots$ of group classes by putting 
$\mathfrak{X}_{1}=\mathfrak{X}$ and choosing 
$\mathfrak{X}_{k+1}$ as the class of all groups whose nonnormal subgroups belong to 
$\mathfrak{X}_{k}$. In particular, if 
$\mathfrak{A}$ is the class of abelian groups, 
$\mathfrak{A}_{2}$ is the class of metahamiltonian groups, that is, groups whose nonnormal subgroups are abelian. The aim of this paper is to study the structure of 
$\mathfrak{X}_{k}$-groups, with special emphasis on the case 
$\mathfrak{X}=\mathfrak{A}$. Among other results, it will be proved that a group has a finite commutator subgroup if and only if it is locally graded and belongs to 
$\mathfrak{A}_{k}$ for some positive integer 
$k$.
