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Journal of the London Mathematical Society 2006 74(2):361-378; doi:10.1112/S0024610706023039
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© The London Mathematical Society 2006

THE NUMBER OF DISTINCT EIGENVALUES OF ELEMENTS IN FINITE LINEAR GROUPS

A. E. Zalesski

School of Mathematics, University of East Anglia Norwich NR4 7TJ, United Kingdom a.zalesskii{at}uea.ac.uk

Received 4 July 2005. Revision received 9 November 2005.

Let G be a finite (non-abelian) irreducible linear subgroup over the complex numbers, and let g be an element of G of prime order p. Suppose that g does not belong to a proper normal subgroup of G. We show that the number of distinct eigenvalues of g can only be p,p – 1,p–2,(p + 1)/2 or (p–1)/2. Moreover, we provide a full classification of such groups G provided that g has at most p–2 distinct eigenvalues.


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