Conjugacy Classes in Sylow p-Subgroups of Finite Chevalley Groups in Bad Characteristic

Abstract

Let U = U(q) be a Sylow p-subgroup of a finite Chevalley group G = G(q). Röhrle and Goodwin in 2009 determined a parameterization of the conjugacy classes of U, for G of small rank when q is a power of a good prime for G. As a consequence they verified that the number k(U) of conjugacy classes of U is given by a polynomial in q with integer coefficients. In the present paper, we consider the case when p is a bad prime for G. Our motivation is to observe how the situation differs between good and bad characteristics. We obtain a parameterization of the conjugacy classes of U, when G has rank less than or equal to 4, and G is not of type F ₄. In these cases we deduce that k(U) is given by a polynomial in q with integer coefficients; this polynomial is different from the polynomial for good primes.

Key Words:

• Chevalley groups
• Conjugacy classes
• Sylow p-subgroups

2010 Mathematics Subject Classification:

• Primary 20G40
• Secondary 20D20
• 20E45

ACKNOWLEDGMENTS

We would like to thank the Universidad de los Andes for supporting a research visit of the second author during which some of this research was completed. We also thank Scott Murray for providing helpful answers to some questions about MAGMA. Further, we are grateful to the referee for useful comments.

Notes

Communicated by D. Macpherson.