Journal of the Mathematical Society of Japan

Some primitive linear groups of prime degree

Ming-chang KANG, Jian-yi SHI, Stephen S. T. YAU, Yung YU, and Ji-ping ZHANG

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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 SL ( n , 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 SL ( 5 , C ) and SL ( 7 , 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 SL ( p , 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 SL ( p , 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.

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

First available in Project Euclid: 6 November 2009

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Primary: 20C15: Ordinary representations and characters

linear groups of prime degree monomial groups


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.

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