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Proceedings of the London Mathematical Society Advance Access published online on April 15, 2008
Proceedings of the London Mathematical Society, doi:10.1112/plms/pdn017
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© 2008 London Mathematical Society

Hall–Higman-type theorems for semisimple elements of finite classical groups

Pham Huu Tiep

Department of Mathematics
University of Florida
Gainesville, FL 32611-8105
USA
tiep@math.ufl.edu

A. E. Zalesskii

School of Mathematics
University of East Anglia
Norwich
NR4 7TJ
United Kingdom

Received 30 August 2007.

Minimum polynomials of semisimple elements of prime power order pa of finite classical groups in (nontrivial) irreducible cross-characteristic representations are studied. In particular, an analogue of the Hall–Higman theorem is established, which shows that the degree of such a polynomial is at least pa–1(p–1), with a few explicit exceptions.

The first author gratefully acknowledges the support of the NSF (grant DMS-0600967), the NSA (grant H98230 [GenBank] -04-0066), and the EPSRC (grant GR/S56511/01). The second author acknowledges the partial support from Levehulme trust (grant EM/2006/0030).

2000 Mathematics Subject Classification 20C15, 20C20, 20C33 (primary), 20G05, 20G40 (secondary).


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