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GROUPS WITH MANY PRONORMAL SUBGROUPS

Part of: Structure and classification of infinite or finite groups Special aspects of infinite or finite groups

Published online by Cambridge University Press:  11 May 2021

MARIA FERRARA Open the ORCID record for MARIA FERRARA [Opens in a new window]  and
MARCO TROMBETTI Open the ORCID record for MARCO TROMBETTI [Opens in a new window]
MARIA FERRARA
Affiliation:
Dipartimento di Matematica e Fisica, Università degli Studi della Campania ‘Luigi Vanvitelli’, Viale Lincoln 5, Caserta, Italy e-mail: [email protected]
MARCO TROMBETTI*
Affiliation:
Dipartimento di Matematica e Applicazioni ‘Renato Caccioppoli’, Università degli Studi di Napoli Federico II, Complesso Universitario Monte S. Angelo, Via Cintia, Napoli, Italy
*
e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

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A subgroup H of a group G is pronormal in G if each of its conjugates $H^g$ in G is conjugate to it in the subgroup $\langle H,H^g\rangle $ ; a group is prohamiltonian if all of its nonabelian subgroups are pronormal. The aim of the paper is to show that a locally soluble group of (regular) cardinality in which all proper uncountable subgroups are prohamiltonian is prohamiltonian. In order to obtain this result, it is proved that the class of prohamiltonian groups is detectable from the behaviour of countable subgroups. Examples are exhibited to show that there are uncountable prohamiltonian groups that do not behave very well. Finally, it is shown that prohamiltonicity can sometimes be detected through the analysis of the finite homomorphic images of a group.

Keywords

prohamiltonian grouppronormal subgroupcountably recognisable group class

MSC classification

Primary: 20E15: Chains and lattices of subgroups, subnormal subgroups
Secondary: 20E25: Local properties 20F16: Solvable groups, supersolvable groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 105 , Issue 1 , February 2022 , pp. 75 - 86
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© Australian Mathematical Publishing Association Inc. 2021

Footnotes

The authors are supported by GNSAGA (INdAM) and are members of the non-profit association ‘Advances in Group Theory and Applications’ (www.advgrouptheory.com). The first author is supported by the project ‘Groups: overlappings between Algebra and Geometry, Logic and Mathematics Education’ (VALERE: VAnviteLli pEr la RicErca).

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