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Dihedral blocks with two simple modules

Author: Frauke M. Bleher
Journal: Proc. Amer. Math. Soc. 138 (2010), 3467-3479
MSC (2010): Primary 20C05; Secondary 16G20
Published electronically: April 27, 2010
MathSciNet review: 2661547
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Abstract: Let $ k$ be an algebraically closed field of characteristic $ 2$, and let $ G$ be a finite group. Suppose $ B$ is a block of $ kG$ with dihedral defect groups such that there are precisely two isomorphism classes of simple $ B$-modules. The description by Erdmann of the quiver and relations of the basic algebra of $ B$ is usually only given up to a certain parameter $ c$ whose value is either 0 or $ 1$. In this article, we show that $ c=0$ if there exists a central extension $ \hat{G}$ of $ G$ by a group of order $ 2$ together with a block $ \hat{B}$ of $ k\hat{G}$ with generalized quaternion defect groups such that $ B$ is contained in the image of $ \hat{B}$ under the natural surjection from $ k\hat{G}$ onto $ kG$. As a special case, we obtain that $ c=0$ if $ G=\mathrm{PGL}_2(\mathbb{F}_q)$ for some odd prime power $ q$ and $ B$ is the principal block of $ k \mathrm{PGL}_2(\mathbb{F}_q)$.

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Additional Information

Frauke M. Bleher
Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242-1419

Keywords: Dihedral defect groups, generalized quaternion defect groups, projective general linear groups
Received by editor(s): July 18, 2009
Received by editor(s) in revised form: August 21, 2009, and January 6, 2010
Published electronically: April 27, 2010
Additional Notes: The author was supported in part by NSF Grant DMS06-51332.
Communicated by: Ted Chinburg
Article copyright: © Copyright 2010 Frauke M. Bleher

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