Centralizers and conjugacy classes in finite classical groups

Abstract

Let $\mathscr{C}$ be a classical group defined over a finite field. We present comprehensive theoretical solutions to the following closely related problems: 1) List a representative for each conjugacy class of $\mathscr{C}$. 2) Given $x \in \mathscr{C}$, describe the centralizer $C_{\mathscr{C}}(x)$ of $x$ in $\mathscr{C}$, by giving its group structure and a generating set. 3) Given $x,y \in \mathscr{C}$, establish whether $x$ and $y$ are conjugate in $\mathscr{C}$ and, if they are, find explicit $z \in \mathscr{C}$ such that $z^{-1}xz = y$. We also formulate practical algorithms to solve these problems and have implemented them in Magma.