## Journal of the Mathematical Society of Japan

### Some primitive linear groups of prime degree

#### Abstract

A classical problem in finite group theory dating back to Jordan, Klein, E. H. Moore, Dickson, Blichfeldt etc. is to determine all finite subgroups in $\mathit{SL} (n,\mbi{C})$ up to conjugation for some small values of $n$. This question is important in group theory as well as in the study of quotient singularities. Some results of Blichfeldt when $n=3,4$ were generalized to the case of finite primitive subgroups of $\mathit{SL} (5,\mbi{C})$ and $\mathit{SL} (7,\mbi{C})$ by Brauer and Wales. The purpose of this article is to consider the following case. Let $p$ be any odd prime number and $G$ be a finite primitive subgroup of $\mathit{SL} (p,\mbi{C})$ containing a non-trivial monomial normal subgroup $H$ so that $H$ has a non-scalar diagonal matrix. We will classify all these groups $G$ up to conjugation in $\mathit{SL} (p,\mbi{C})$ by exhibiting the generators of $G$ and representing $G$ as some group extensions. In particular, see the Appendix for a list of these subgroups when $p=5$ or 7.

#### Article information

Source
J. Math. Soc. Japan, Volume 61, Number 4 (2009), 1013-1070.

Dates
First available in Project Euclid: 6 November 2009

https://projecteuclid.org/euclid.jmsj/1257520499

Digital Object Identifier
doi:10.2969/jmsj/06141013

Mathematical Reviews number (MathSciNet)
MR2588503

Zentralblatt MATH identifier
1211.20040

Subjects
Primary: 20C15: Ordinary representations and characters

#### Citation

KANG, Ming-chang; ZHANG, Ji-ping; SHI, Jian-yi; YU, Yung; YAU, Stephen S. T. Some primitive linear groups of prime degree. J. Math. Soc. Japan 61 (2009), no. 4, 1013--1070. doi:10.2969/jmsj/06141013. https://projecteuclid.org/euclid.jmsj/1257520499

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