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Blocks with central product defect group $ D_{2^n}\ast C_{2^m}$

Author: Benjamin Sambale
Journal: Proc. Amer. Math. Soc. 141 (2013), 4057-4069
MSC (2010): Primary 20C15, 20C20
Published electronically: August 14, 2013
MathSciNet review: 3105851
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Abstract: We determine the numerical invariants of blocks with defect group $ D_{2^n}\ast C_{2^m}\cong Q_{2^n}\ast C_{2^m}$ (central product), where $ n\ge 3$ and $ m\ge 2$. As a consequence, we prove Brauer's $ k(B)$-conjecture, Olsson's conjecture (and more generally Eaton's conjecture), Brauer's height zero conjecture, the Alperin-McKay conjecture, Alperin's weight conjecture and Robinson's ordinary weight conjecture for these blocks. Moreover, we show that the gluing problem has a unique solution in this case. This paper continues B. Sambale, Blocks with defect group $ D_{2^n}\times C_{2^m}$, J. Pure Appl. Algebra 216 (2012), 119-125.

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

Keywords: $2$-blocks, dihedral defect groups, Alperin's weight conjecture, ordinary weight conjecture
Received by editor(s): June 8, 2011
Received by editor(s) in revised form: February 1, 2012
Published electronically: August 14, 2013
Additional Notes: This work was partly supported by the Deutsche Forschungsgemeinschaft
Communicated by: Birge Huisgen-Zimmermann
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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