The binary actions of simple groups of Lie type of characteristic 2

Gill, Nick; Guillot, Pierre and Liebeck, Martin (2024). The binary actions of simple groups of Lie type of characteristic 2. Pacific Journal of Mathematics (In press).

Abstract

Let $\mathcal{C}$ be a conjugacy class of involutions in a group $G$. We study the graph $\Gamma(\mathcal{C})$ whose vertices are elements of $\mathcal{C}$ with $g,h\in\mathcal{C}$ connected by an edge if and only if $gh\in\mathcal{C}$. For $t\in \mathcal{C}$, we define the \emph{component group} of $t$ to be the subgroup of $G$ generated by all vertices in $\Gamma(\mathcal{C})$ that lie in the connected component of the graph that contains $t$.

We classify the component groups of all involutions in simple groups of Lie type over a field of characteristic $2$. We use this classification to partially classify the transitive binary actions of the simple groups of Lie type over a field of characteristic $2$ for which a point stabilizer has even order. The classification is complete unless the simple group in question is a symplectic or unitary group.

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