Regular orbits of sporadic simple 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 nontrivial $g\in G/Z(G)$, we compute the minimal number of $T$conjugates of $g$ generating $\langle T,g\rangle$.
 Publication:

arXiv eprints
 Pub Date:
 January 2018
 arXiv:
 arXiv:1801.04561
 Bibcode:
 2018arXiv180104561F
 Keywords:

 Mathematics  Representation Theory;
 Mathematics  Group Theory;
 20C20;
 20C34;
 20C40;
 20B15
 EPrint:
 17 pages, shortened proof plus new result (Theorem 1.3)