Orbit equivalence and permutation groups defined by unordered relations

Dalla Volta, Francesca ; Siemons, Johannes

Journal of Algebraic Combinatorics, Tome 35 (2012) no. 4, pp. 547-564.

Résumé

Summary: 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 $G( R)$ mathcalG$(R)$ be the group of all permutations on $\Omega $ that preserve $R$, that is $g$ belongs to $G( R)$ mathcalG$(R)$ if and only if $x\in R$ implies $x ^{ g }\in R$. We are interested in permutation groups which can be represented as $G= G( R)$ G=mathcalG$(R)$ 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 $\neq Alt( \Omega )$ and of degree $\geq 11$ is a relation group. The same is true for many classes of finite imprimitive groups, and we give general conditions on the size of blocks of imprimitivity, 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.

Détail

Keywords: keywords group invariant relations, regular sets, orbit closure, automorphism groups of set systems