Recognition of finite exceptional groups of Lie type

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 $G otcong {}^3 D_4(q)$ with $q$ even, then the algorithm runs in polynomial time, subject to the existence of a discrete log oracle.

Some word maps that are non-surjective on infinitely many finite simple groups

We provide the first examples of words in the free group of rank 2 which are not proper powers and for which the corresponding word maps are non-surjective on an infinite family of finite non-abelian simple groups.