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Groups where all the irreducible characters are super-monomial


Author: Mark L. Lewis
Journal: Proc. Amer. Math. Soc. 138 (2010), 9-16
MSC (2000): Primary 20C15
DOI: https://doi.org/10.1090/S0002-9939-09-10059-X
Published electronically: August 13, 2009
MathSciNet review: 2550165
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Abstract: Isaacs has defined a character to be super-monomial if every primitive character inducing it is linear. Isaacs has conjectured that if $ G$ is an $ M$-group with odd order, then every irreducible character is super-monomial. We prove that the conjecture is true if $ G$ is an $ M$-group of odd order where every irreducible character is a $ \{p\}$-lift for some prime $ p$. We say that a group where every irreducible character is super-monomial is a super $ M$-group. We use our results to find an example of a super $ M$-group that has a subgroup that is not a super $ M$-group.


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Additional Information

Mark L. Lewis
Affiliation: Department of Mathematical Sciences, Kent State University, Kent, Ohio 44242
Email: lewis@math.kent.edu

DOI: https://doi.org/10.1090/S0002-9939-09-10059-X
Keywords: $\pi $-partial characters, lifts, $M$-groups, super monomial characters
Received by editor(s): December 15, 2008
Published electronically: August 13, 2009
Communicated by: Jonathan I. Hall
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.