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FINITE GROUPS WHOSE NONCENTRAL COMMUTING ELEMENTS HAVE CENTRALIZERS OF EQUAL SIZE

Part of: Representation theory of groups Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  07 July 2010

SILVIO DOLFI ,
MARCEL HERZOG  and
ENRICO JABARA
SILVIO DOLFI
Affiliation:
Dipartimento di Matematica U. Dini, Università degli Studi di Firenze, viale Morgagni 67/a, 50134 Firenze, Italy (email: [email protected])
MARCEL HERZOG*
Affiliation:
Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel (email: [email protected])
ENRICO JABARA
Affiliation:
Dipartimento di Matematica Applicata, Università di Ca’Foscari, 30123 Venezia, Italy (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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A finite group is called a CH-group if for every x,y∈G∖Z(G), xy=yx implies that . Applying results of Schmidt [‘Zentralisatorverbände endlicher Gruppen’, Rend. Sem. Mat. Univ. Padova44 (1970), 97–131] and Rebmann [‘F-Gruppen’, Arch. Math. 22 (1971), 225–230] concerning CA-groups and F-groups, the structure of CH-groups is determined, up to that of CH-groups of prime-power order. Upper bounds are found for the derived length of nilpotent and solvable CH-groups.

Keywords

centralizersorder of centralizersF-groupsCA-groups

MSC classification

Secondary: 20E34: General structure theorems 20D60: Arithmetic and combinatorial problems 20D15: Nilpotent groups, $p$-groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 82 , Issue 2 , October 2010 , pp. 293 - 304
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

The first and the third authors were partially supported by the MIUR project ‘Teoria dei gruppi e applicazioni’.

References

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