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

DOI:
https://doi.org/10.1090/S0002-9939-10-10402-X

Published electronically:
April 27, 2010

MathSciNet review:
2661547

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be an algebraically closed field of characteristic , and let be a finite group. Suppose is a block of with dihedral defect groups such that there are precisely two isomorphism classes of simple -modules. The description by Erdmann of the quiver and relations of the basic algebra of is usually only given up to a certain parameter whose value is either 0 or . In this article, we show that if there exists a central extension of by a group of order together with a block of with generalized quaternion defect groups such that is contained in the image of under the natural surjection from onto . As a special case, we obtain that if for some odd prime power and is the principal block of .

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Additional Information

**Frauke M. Bleher**

Affiliation:
Department of Mathematics, University of Iowa, Iowa City, Iowa 52242-1419

Email:
fbleher@math.uiowa.edu

DOI:
https://doi.org/10.1090/S0002-9939-10-10402-X

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