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The Alperin-McKay Conjecture for metacyclic, minimal non-abelian defect groups

Author: Benjamin Sambale
Journal: Proc. Amer. Math. Soc. 143 (2015), 4291-4304
MSC (2010): Primary 20C15, 20C20
Published electronically: April 1, 2015
MathSciNet review: 3373928
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Abstract: We prove the Alperin-McKay Conjecture for all $ p$-blocks of finite groups with metacyclic, minimal non-abelian defect groups. These are precisely the metacyclic groups whose derived subgroup have order $ p$. In the special case $ p=3$, we also verify Alperin's Weight Conjecture for these defect groups. Moreover, in case $ p=5$ we do the same for the non-abelian defect groups $ C_{25}\rtimes C_{5^n}$. The proofs do not rely on the classification of the finite simple groups.

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Benjamin Sambale
Affiliation: Institut für Mathematik, Friedrich-Schiller-Universität, 07743 Jena, Germany

Keywords: Alperin-McKay Conjecture, metacyclic defect groups
Received by editor(s): April 13, 2014
Received by editor(s) in revised form: July 16, 2014
Published electronically: April 1, 2015
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2015 American Mathematical Society

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