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GROUPS WHOSE NONNORMAL SUBGROUPS ARE METAHAMILTONIAN

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  08 October 2019

DARIO ESPOSITO Open the ORCID record for DARIO ESPOSITO [Opens in a new window] ,
FRANCESCO DE GIOVANNI Open the ORCID record for FRANCESCO DE GIOVANNI [Opens in a new window]  and
MARCO TROMBETTI Open the ORCID record for MARCO TROMBETTI [Opens in a new window]
DARIO ESPOSITO
Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Napoli, Italy email [email protected]
FRANCESCO DE GIOVANNI*
Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Napoli, Italy email [email protected]
MARCO TROMBETTI
Affiliation:
Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Napoli, Italy email [email protected]
*
✉[email protected]
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Abstract

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 $k$.

Keywords

metahamiltonian groupk-hamiltonian groupfinite-by-abelian group

MSC classification

Primary: 20F24: FC-groups and their generalizations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 102 , Issue 1 , August 2020 , pp. 96 - 103
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Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

The authors are supported by GNSAGA (INdAM); the last two authors are members of AGTA—Advances in Group Theory and Applications (www.advgrouptheory.com).

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