A reduction theorem for primitive binary permutation groups

Abstract

A permutation group $(X,G)$ is said to be binary, or of relational complexity 2, if, for all $n$, the orbits of $G$ (acting diagonally) on $X^2$ determine the orbits of $G$ on $X^n$ in the following sense: for all $\bar {x},\bar {y} \in X^n$, $\bar {x}$ and $\bar {y}$ are $G$-conjugate if and only if every pair of entries from $\bar {x}$ is $G$-conjugate to the corresponding pair from $\bar {y}$. Cherlin has conjectured that the only finite primitive binary permutation groups are $S-n$, groups of prime order, and affine orthogonal groups $V\rtimes O(V),$ where $V$ is a vector space equipped with an anisotropic quadratic form; recently, he succeeded in establishing the conjecture for those groups with an abelian socle. In this note, we show that what remains of the conjecture reduces, via the O'Nan-Scott Theorem, to groups with a nonabelian simple socle.