A reduction theorem for primitive binary permutation groups

TL;DR

This paper proves a reduction step for Cherlin's conjecture on finite primitive binary permutation groups, showing that the remaining cases can be reduced to groups with a nonabelian simple socle using the O'Nan-Scott Theorem.

Contribution

It reduces the classification problem of primitive binary groups to the case with a nonabelian simple socle, simplifying the overall conjecture.

Findings

Reduces the classification of primitive binary groups to nonabelian simple socle cases.

Uses the O'Nan-Scott Theorem to achieve this reduction.

Progress towards resolving Cherlin's conjecture.

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.