CONJUGACY CLASSES IN SYLOW p-SUBGROUPS OF FINITE CHEVALLEY GROUPS IN BAD CHARACTERISTIC

Let U=U(q) be a Sylow p-subgroup of a finite Chevalley group G=G(q). Rohrle 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.