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GROUPS WITH COMMUTING POWERS

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

Published online by Cambridge University Press:  13 March 2009

ROLF BRANDL
ROLF BRANDL*
Affiliation:
Mathematisches Institut, Am Hubland 12, 97074 Würzburg, Germany
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Abstract

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A group G satisfies the second Engel condition [X,Y,Y ]=1 if and only if x commutes with xy, for all x,y∈G. This paper considers the generalization of this condition to groups G such that, for fixed positive integers r and s, xr commutes with (xs)y for all x,y∈G. Various general bounds are proved for the structure of groups in the corresponding variety, defined by the law [Xr,(Xs)Y]=1.

Keywords

two-EngelBell groupsvariety of groups

MSC classification

Secondary: 20F19: Generalizations of solvable and nilpotent groups 20F12: Commutator calculus 20F45: Engel conditions 20D60: Arithmetic and combinatorial problems 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $pi$-length, ranks
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 79 , Issue 2 , April 2009 , pp. 343 - 351
Copyright
Copyright © Australian Mathematical Society 2009

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