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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Isaacs has defined a character to be super-monomial if every primitive character inducing it is linear. Isaacs has conjectured that if is an -group with odd order, then every irreducible character is super-monomial. We prove that the conjecture is true if is an -group of odd order where every irreducible character is a -lift for some prime . We say that a group where every irreducible character is super-monomial is a super -group. We use our results to find an example of a super -group that has a subgroup that is not a super -group.

**1.**Bertram Huppert,*Character theory of finite groups*, De Gruyter Expositions in Mathematics, vol. 25, Walter de Gruyter & Co., Berlin, 1998. MR**1645304****2.**I. Martin Isaacs,*Character theory of finite groups*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. Pure and Applied Mathematics, No. 69. MR**0460423****3.**I. M. Isaacs,*Characters of 𝜋-separable groups*, J. Algebra**86**(1984), no. 1, 98–128. MR**727371**, https://doi.org/10.1016/0021-8693(84)90058-9**4.**I. M. Isaacs,*Induction and restriction of 𝜋-special characters*, Canad. J. Math.**38**(1986), no. 3, 576–604. MR**845666**, https://doi.org/10.4153/CJM-1986-029-5**5.**I. M. Isaacs,*Characters of solvable groups*, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986) Proc. Sympos. Pure Math., vol. 47, Amer. Math. Soc., Providence, RI, 1987, pp. 103–109. MR**933354****6.**I. M. Isaacs,*The 𝜋-character theory of solvable groups*, J. Austral. Math. Soc. Ser. A**57**(1994), no. 1, 81–102. MR**1279288****7.**I. Martin Isaacs,*Algebra*, Brooks/Cole Publishing Co., Pacific Grove, CA, 1994. A graduate course. MR**1276273****8.**I. M. Isaacs,*Characters and sets of primes for solvable groups*, Finite and locally finite groups (Istanbul, 1994) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 471, Kluwer Acad. Publ., Dordrecht, 1995, pp. 347–376. MR**1362816**, https://doi.org/10.1007/978-94-011-0329-9_13**9.**I. M. Isaacs,*Induction and restriction of 𝜋-partial characters and their lifts*, Canad. J. Math.**48**(1996), no. 6, 1210–1223. MR**1426901**, https://doi.org/10.4153/CJM-1996-064-9**10.**Olaf Manz and Thomas R. Wolf,*Representations of solvable groups*, London Mathematical Society Lecture Note Series, vol. 185, Cambridge University Press, Cambridge, 1993. MR**1261638****11.**Gabriel Navarro,*New properties of the 𝜋-special characters*, J. Algebra**187**(1997), no. 1, 203–213. MR**1425567**, https://doi.org/10.1006/jabr.1997.6798

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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.