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The group of endotrivial modules for the symmetric and alternating groups

Published online by Cambridge University Press:  12 January 2010

Jon F. Carlson
Affiliation:
Department of Mathematics, University of Georgia, Athens, GA 30602, USA
David J. Hemmer
Affiliation:
Department of Mathematics, University at Buffalo SUNY, 244 Mathematics Building, Buffalo, NY 14260, USA, Email: ([email protected])
Nadia Mazza
Affiliation:
Department of Mathematical Sciences, University of Aberdeen, Aberdeen AB24 3UE, UK
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Abstract

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We complete a classification of the groups of endotrivial modules for the modular group algebras of symmetric groups and alternating groups. We show that, for np2, the torsion subgroup of the group of endotrivial modules for the symmetric groups is generated by the sign representation. The torsion subgroup is trivial for the alternating groups. The torsion-free part of the group is free abelian of rank 1 if np2 + p and has rank 2 if p2n < p2 + p. This completes the work begun earlier by Carlson, Mazza and Nakano.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2010