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Schur indices and commutators in supersolvable groups

Schur indices and commutators in supersolvable groups J. Group Theory 11 (2008), 691­695 DOI 10.1515/JGT.2008.044 ( de Gruyter 2008 Michael Geline (Communicated by N. Boston) If G is a finite group and w A IrrðGÞ, then the Schur index of w over the field F of characteristic zero is the smallest multiplicity with which w can appear in a character a¤orded by an FG-module. This multiplicity is denoted by mF ðwÞ. If mF ðwÞ is divisible by the prime p, a generalization of Brauer's induction theorem implies that G contains a subgroup H which has an irreducible character z such that p does not divide ½wH ; zjF ðw; zÞ : F ðwÞj. From this, basic properties of the Schur index imply that the p-part of mF ðzÞ is at least as large as that of mF ðwÞ. The strength of the theorem is that H can be chosen to be the semidirect product of a p-group acting on a cyclic p 0 -group. Such groups are called p-hyperelementary groups. One then seeks an irreducible character z0 of some proper subgroup H0 of H=KerðzÞ such that p does not divide ½zH0 ; z0 jF ðz; z0 Þ : F ðzÞj, and asks about the p-hyperelementary http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Group Theory de Gruyter

Schur indices and commutators in supersolvable groups

Geline, Michael
Journal of Group Theory , Volume 11 (5) – Sep 1, 2008
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References (3)

Publisher
de Gruyter
Copyright
Copyright © 2008 by the
ISSN
1433-5883
eISSN
1435-4446
DOI
10.1515/JGT.2008.044
Publisher site
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Abstract

J. Group Theory 11 (2008), 691­695 DOI 10.1515/JGT.2008.044 ( de Gruyter 2008 Michael Geline (Communicated by N. Boston) If G is a finite group and w A IrrðGÞ, then the Schur index of w over the field F of characteristic zero is the smallest multiplicity with which w can appear in a character a¤orded by an FG-module. This multiplicity is denoted by mF ðwÞ. If mF ðwÞ is divisible by the prime p, a generalization of Brauer's induction theorem implies that G contains a subgroup H which has an irreducible character z such that p does not divide ½wH ; zjF ðw; zÞ : F ðwÞj. From this, basic properties of the Schur index imply that the p-part of mF ðzÞ is at least as large as that of mF ðwÞ. The strength of the theorem is that H can be chosen to be the semidirect product of a p-group acting on a cyclic p 0 -group. Such groups are called p-hyperelementary groups. One then seeks an irreducible character z0 of some proper subgroup H0 of H=KerðzÞ such that p does not divide ½zH0 ; z0 jF ðz; z0 Þ : F ðzÞj, and asks about the p-hyperelementary

Journal

Journal of Group Theory – de Gruyter

Published: Sep 1, 2008

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