Recognition of finite exceptional groups of Lie type

TL;DR

This paper introduces a Las Vegas algorithm for recognizing finite exceptional groups of Lie type over finite fields, providing an efficient method to identify and construct isomorphisms with standard groups, with some exceptions.

Contribution

It presents a polynomial-time recognition algorithm for most finite exceptional Lie type groups, advancing computational group theory methods.

Findings

Algorithm successfully recognizes most exceptional Lie groups

Runs in polynomial time given a discrete log oracle

Handles a broad class of finite simple groups of Lie type

Abstract

Let $q$ be a prime power and let $G$ be an absolutely irreducible subgroup of $GL_d(F)$, where $F$ is a finite field of the same characteristic as $\F_q$, the field of $q$ elements. Assume that $G \cong G(q)$, a quasisimple group of exceptional Lie type over $\F_q$ which is neither a Suzuki nor a Ree group. We present a Las Vegas algorithm that constructs an isomorphism from $G$ to the standard copy of $G(q)$. If $G \not\cong {}^3 D_4(q)$ with $q$ even, then the algorithm runs in polynomial time, subject to the existence of a discrete log oracle.