Regular orbits of sporadic simple groups

TL;DR

This paper classifies when sporadic simple groups acting on certain modules lack regular orbits, linking group actions to base sizes in permutation representations, and provides a detailed analysis of generating conjugates.

Contribution

It provides a complete classification of pairs (G,V) where G, a covering of an almost simple sporadic group, has no regular orbit on V, and determines the minimal base size.

Findings

Identifies all pairs (G,V) with no regular orbit.

Determines minimal base sizes for these actions.

Analyzes conjugate generating sets for sporadic groups.

Abstract

Given a finite group $G$ and a faithful irreducible $FG$-module $V$ where $F$ has prime order, does $G$ have a regular orbit on $V$? This problem is equivalent to determining which primitive permutation groups of affine type have a base of size 2. Let $G$ be a covering group of an almost simple group whose socle $T$ is sporadic, and let $V$ be a faithful irreducible $FG$-module where $F$ has prime order dividing $|G|$. We classify the pairs $(G,V)$ for which $G$ has no regular orbit on $V$, and determine the minimal base size of $G$ in its action on $V$. To obtain this classification, for each non-trivial $g\in G/Z(G)$, we compute the minimal number of $T$-conjugates of $g$ generating $\langle T,g\rangle$.