Conjugacy in finite classical groups

Giovanni De Franceschi, Martin W. Liebeck, E. A. O'Brien

Published 2024-01-15Version 1

Let $G$ be a classical group defined over a finite field. We consider the following fundamental problems concerning conjugacy in $G$: 1. List a representative for each conjugacy class of $G$. 2. Given $x \in G$, describe the centralizer of $x$ in $G$, by giving its group structure and a generating set. 3. Given $x,y \in G$, establish whether $x$ and $y$ are conjugate in $G$ and, if so, then find explicit $z \in G$ such that $z^{-1}xz = y$. We present comprehensive theoretical solutions to all three problems, and use our solutions to formulate practical algorithms. In parallel to our theoretical work, we have developed in Magma complete implementations of our algorithms. They form a critical component of various general algorithms in computational group theory - for example, computing character tables and solving conjugacy problems in arbitrary finite groups.