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ON GROUPS WITH FINITE CONJUGACY CLASSES IN A VERBAL SUBGROUP

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

Published online by Cambridge University Press:  08 June 2017

COSTANTINO DELIZIA ,
PAVEL SHUMYATSKY  and
ANTONIO TORTORA Open the ORCID record for ANTONIO TORTORA [Opens in a new window]
COSTANTINO DELIZIA
Affiliation:
Dipartimento di Matematica, Università di Salerno, Via Giovanni Paolo II, 132 - 84084 - Fisciano (SA), Italy email cdelizia@unisa.it
PAVEL SHUMYATSKY
Affiliation:
Department of Mathematics, University of Brasilia, Brasilia-DF, 70910-900, Brazil email pavel@unb.br
ANTONIO TORTORA*
Affiliation:
Dipartimento di Matematica, Università di Salerno, Via Giovanni Paolo II, 132 - 84084 - Fisciano (SA), Italy email antortora@unisa.it
*
✉ antortora@unisa.it
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Abstract

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Let $w$ be a group-word. For a group $G$ , let $G_{w}$ denote the set of all $w$ -values in $G$ and let $w(G)$ denote the verbal subgroup of $G$ corresponding to $w$ . The group $G$ is an $FC(w)$ -group if the set of conjugates $x^{G_{w}}$ is finite for all $x\in G$ . It is known that if $w$ is a concise word, then $G$ is an $FC(w)$ -group if and only if $w(G)$ is $FC$ -embedded in $G$ , that is, the conjugacy class $x^{w(G)}$ is finite for all $x\in G$ . There are examples showing that this is no longer true if $w$ is not concise. In the present paper, for an arbitrary word $w$ , we show that if $G$ is an $FC(w)$ -group, then the commutator subgroup $w(G)^{\prime }$ is $FC$ -embedded in $G$ . We also establish the analogous result for $BFC(w)$ -groups, that is, groups in which the sets $x^{G_{w}}$ are boundedly finite.

Keywords

conjugacy classFC-groupverbal subgroup

MSC classification

Primary: 20E45: Conjugacy classes
Secondary: 20F24: FC-groups and their generalizations

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 96 , Issue 3 , December 2017 , pp. 429 - 437
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

Footnotes

The second author was supported by the ‘National Group for Algebraic and Geometric Structures and their Applications’ (GNSAGA – INdAM) and FAPDF-Brazil.

References

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