Conjugacy classes in Sylow p-subgroups of finite Chevalley groups in bad characteristic

TL;DR

This paper studies the conjugacy classes of Sylow p-subgroups in finite Chevalley groups when p is a bad prime, providing parameterizations and polynomial formulas for the number of classes in low-rank cases.

Contribution

It extends previous work by parameterizing conjugacy classes and deriving polynomial formulas for the number of classes in bad prime cases for groups of rank ≤ 4, excluding type F4.

Findings

Parameterization of conjugacy classes for rank ≤ 4 groups in bad characteristic.

Number of conjugacy classes is given by a polynomial in q with integer coefficients.

Polynomial differs from the good prime case.

Abstract

Let $U = \mathbf U(q)$ be a Sylow $p$-subgroup of a finite Chevalley group $G = \mathbf G(q)$. In [GR}] R\"ohrle and the second author determined a parameterization of the conjugacy classes of $U$, for $\mathbf G$ of small rank when $q$ is a power of a good prime for $\mathbf G$. 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 $\mathbf G$. We obtain a parameterization of the conjugacy classes of $U$, when $\mathbf G$ has rank less than or equal to 4, and $\mathbf G$ 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.