Orbit Coherence in Permutation Groups

Orbit equivalence and permutation groups defined by unordered relations

For a set Ω an unordered relation on Ω is a family R of subsets of Ω . If R is such a relation we let G(R) be the group of all permutations on Ω that preserve R, that is g belongs to G(R) if and only if x ∈R implies x ∈R. We are interested in permutation groups which can be represented as G= G(R) for a suitable unordered relation R on Ω . When this is the case, we say that G is defined by the r...

Computation in permutation groups: ounting and randomly sampling orbits

Symmetry-Breaking Formulas for Groups with Bounded Orbit Projections

This paper consider the problem of generating symmetry-breaking formulas for permutation groups acting on n points, where the projection of the group into each orbit is bounded by a polynomial in n. We show that the lex-leader formula, which is satisfied by the lexicographic leader of each set of symmetrical points in the search space, can be generated in polynomial time (and space). Our constr...

Infinite Permutation Groups in Enumeration and Model Theory

A permutation group G on a set Q has a natural action on Q for each natural number n. The group is called oligomorphic if it has only finitely many orbits on Q for all «GN. (The term means "few shapes". Typically our permutation groups are groups of automorphisms of structures of some kind; oligomorphy implies that the structure has only finitely many non-isomorphic w-element substructures for ...

Finite permutation groups of rank 3

By the rank of a transitive permutation group we mean the number of orbits of the stabilizer of a point thus rank 2 means multiple transitivity. Interest is drawn to the simply transitive groups of "small" rank > 2 by the fact that every known finite simple group admits a representation as a primitive group of rank at most 5 while not all of these groups have doubly transitive representations. ...