Repeated and final commutators in group actions
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- by I. M. Isaacs and Ulrich Meierfrankenfeld
- Proc. Amer. Math. Soc. 140 (2012), 3777-3783
- DOI: https://doi.org/10.1090/S0002-9939-2012-11228-6
- Published electronically: March 14, 2012
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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 $[G,A]$.References
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Bibliographic 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
- 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
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2944718