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ON THE RANK OF A VERBAL SUBGROUP OF A FINITE GROUP

Part of: Representation theory of groups Special aspects of infinite or finite groups

Published online by Cambridge University Press:  12 May 2021

ELOISA DETOMI ,
MARTA MORIGI Open the ORCID record for MARTA MORIGI [Opens in a new window]  and
PAVEL SHUMYATSKY
ELOISA DETOMI
Affiliation:
Dipartimento di Ingegneria dell’Informazione, Università di Padova, Via G. Gradenigo 6/B, 35121Padova, Italy e-mail: eloisa.detomi@unipd.it
MARTA MORIGI*
Affiliation:
Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40126Bologna, Italy
PAVEL SHUMYATSKY
Affiliation:
Department of Mathematics, University of Brasilia, Brasilia-DF, 70910-900, Brazil e-mail: pavel@unb.br
*
e-mail: marta.morigi@unibo.it
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Abstract

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We show that if w is a multilinear commutator word and G a finite group in which every metanilpotent subgroup generated by w-values is of rank at most r, then the rank of the verbal subgroup $w(G)$ is bounded in terms of r and w only. In the case where G is soluble, we obtain a better result: if G is a finite soluble group in which every nilpotent subgroup generated by w-values is of rank at most r, then the rank of $w(G)$ is at most $r+1$ .

Keywords

multilinear commutator wordsrankverbal subgroup

MSC classification

Primary: 20D20: Sylow subgroups, Sylow properties, $pi$-groups, $pi$-structure
Secondary: 20F12: Commutator calculus
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 113 , Issue 2 , October 2022 , pp. 145 - 159
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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 (https://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
© The Author(s), 2021. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Footnotes

Communicated by Ben Martin

The first and second authors are members of GNSAGA (Indam). The third author was partially supported by FAPDF and CNPq.

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