Cherlin’s conjecture for sporadic simple groups

Abstract

We prove Cherlin's conjecture, concerning binary primitive permutation groups, for those groups with socle isomorphic to a sporadic simple group.

Classical Groups

“…In [34,Lemma 2.5], the hypothesis actually requires that h and gh are conjugate to g via G; however the same proof yields the conclusion that G is not binary under the weaker assuption that h and gh are conjugate to g in G, as stated in the lemma. We will need this strengthening in what follows.…”

“…Most of the results in this section were first written down in [34,45,46]. All of these papers were focused on showing that certain group actions are not binary, hence the lemmas we present here tend to yield lower bounds for relational complexity.…”

“…The next two lemmas which are taken from [34] are based on Example 1.6.14. In both cases, the given assumptions on the permutation group G are enough to conclude that a strongly non-binary subset of the type described in Example 1.6.14 must exist.…”

“…In [46], Conjecture 1.2 was proved for groups with socle a simple alternating group; in [34], Conjecture 1.2 was proved for groups with socle a sporadic simple group. In this monograph we deal with the remaining family.…”

“…A special case of Theorem 1.3 has already appeared in the literature; in [34], the theorem is proved for the case where G is almost simple with socle a finite group of Lie type of rank 1.…”