Orbit closed permutation groups, relation groups, and simple groups

Abstract

A permutation group G on a set $$\Omega $$ Ω is called orbit closed if every permutation of $$\Omega $$ Ω preserving the orbits of G in its action on the power set $$P(\Omega )$$ P ( Ω ) belongs to G. It is called a relation group if there exists a family $$R \subseteq P(\Omega )$$ R ⊆ P ( Ω ) such that G is the group of all permutations preserving R. We prove that if a finite orbit closed permutation group G is simple, or is a subgroup of a simple group, then it is a relation group. This result justifies our general conjecture that with a few exceptions every finite orbit closed group is a relation group. To obtain the result, we prove that most of the finite simple permutation groups are relation groups. We also obtain a complete description of those finite simple permutation groups that have regular sets, and prove that (with one exception) if a finite simple permutation group G is a relation group, then every subgroup of G is a relation group.

Symmetry groups for social preference functions

“…The only known group in BGR \ BGR(2) is the Klein group K but it is unknown even if BGR \ BGR(2) is finite or not. For the moment that set is a pure mystery, which to be resilient also to some massive recent attacks ( [12], [13], [14]). Remarkably, in [14] the authors characterize the finite simple groups in BGR (2).…”

“…For the moment that set is a pure mystery, which to be resilient also to some massive recent attacks ( [12], [13], [14]). Remarkably, in [14] the authors characterize the finite simple groups in BGR (2). We emphasize that those recent results seem to confirm that BGR \ BGR(2) = {K}.…”

“…While dealing with this problem in social choice theory, we immediately realized a main surprisingly connection between the anonymity problem for spfs and the representability by Boolean functions, a topic having a long history (see, for instance, [5], [11], [12], [13], [7], [8], [14]) and arising in the study of parallel complexity of formal languages. Surely our methods and results impact on classic representability questions, and we give some contribution throughout the paper.…”