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Repeated and final commutators in group actions


Authors: I. M. Isaacs and Ulrich Meierfrankenfeld
Journal: Proc. Amer. Math. Soc. 140 (2012), 3777-3783
MSC (2010): Primary 20D45, 20D35, 20D30
DOI: https://doi.org/10.1090/S0002-9939-2012-11228-6
Published electronically: March 14, 2012
MathSciNet review: 2944718
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Abstract: Let $ G$ be a finite group and suppose that $ A$ acts via automorphisms on $ G$. The repeated commutators are the subgroups $ [G,A,A,\ldots ,A]$, where there is some positive number of commutations by $ A$, and the final commutator is the smallest of these repeated commutators. We show that if $ [G,A]$ is nilpotent, then the final commutator is normal in $ G$. Also, in general, if $ K$ is an arbitrary repeated commutator and $ P$ is the permutation group induced by the action of $ A$ on the left cosets of $ K$ in $ G$, we relate the structure of $ P$ to the structure of $ [G,A]$.


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Additional Information

I. M. Isaacs
Affiliation: Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, Wisconsin 53706
Email: isaacs@math.wisc.edu

Ulrich Meierfrankenfeld
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: meier@math.msu.edu

DOI: https://doi.org/10.1090/S0002-9939-2012-11228-6
Keywords: Automorphism group, commutator, nilpotent group, SHP-class.
Received by editor(s): July 21, 2010
Received by editor(s) in revised form: May 3, 2011
Published electronically: March 14, 2012
Communicated by: Jonathan I. Hall
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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