On Orbit Equivalence and Permutation groups defined by unordered relations

TL;DR

This paper investigates permutation groups that can be represented by unordered relations, proving that many primitive and imprimitive groups are relation groups and thus orbit closed, expanding understanding of their structure.

Contribution

It establishes that all primitive permutation groups (except the alternating group) of degree at least 11 are relation groups, and provides conditions for certain imprimitive groups to be relation groups.

Findings

Primitive groups (not alternating) ≥11 are relation groups.

Many imprimitive groups are relation groups under certain conditions.

Relation groups are closely linked to orbit closure properties.

Abstract

For a set $\Omega$ an unordered relation on $\Omega$ is a family R of subsets of $\Omega.$ If R is such a relation we let G(R) be the group of all permutations on $\Omega$ that preserves R, that is g belongs to G(R) if and only if x in R implies x^{g}\in R. We are interested in permutation groups which can be represented as G=G(R) for a suitable unordered relation R on $\Omega.$ When this is the case, we say that G is defined by the relation R, or that G is a relation group. We prove that a primitive permutation group different from the Alternating Group and of degree bigger or equal to 11 is a relation groups. The same is true for many classes of finite imprimitive groups, and we give general conditions on the size of blocks of imprmitivity, and the groups induced on such blocks, which guarantee that the group is defined by a relation. This property is closely connected to the orbit closure of permutation groups. Since relation groups are orbit closed the results here imply that many classes of imprimitive permutation groups are orbit closed.