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Finite Linear Groups of Degree Six

Published online by Cambridge University Press:  20 November 2018

J. H. Lindsey II
J. H. Lindsey II*
Affiliation:
Northern Illinois University, Dekalb, Illinois
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In this paper we classify finite groups G with a faithful, quasiprimitive (see Notation), unimodular representation X with character χ of degree six over the complex number field. There are three gaps in the proof which are filled in by [16; 17]. These gaps concern existence and uniqueness of simple, projective, complex linear groups of order 604800, |LF(3, 4)|, and |PSL4(3)|. By [19], X is a tensor product of a 2-dimensional and a 3-dimensional group, or a subgroup thereof, or X corresponds to a projective representation of a simple group, possibly extended by some automorphisms. The tensor product case is discussed in section 10. Otherwise, we assume that G/Z(G) is simple. We discuss which automorphisms of G/Z(G) extend the representation X (that is, lift to the central extension G and fix the character corresponding to X) just after we find X(G).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 5 , October 1971 , pp. 771 - 790
Copyright
Copyright © Canadian Mathematical Society 1971

Footnotes

The author wishes to thank Professor Brauer for his help and encouragement, and for suggesting many of the techniques.

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