of a square is transitive on the vertices. Primitive actions A permutation group G acting transitively on a non-empty finite set M is imprimitive if there is some nontrivial set partition of M that is preserved by the action of G, where "nontrivial" means that the partition isn't the partition into singleton sets nor the partition with only one part. Otherwise, if G is transitive but does not preserve any nontrivial partition of M, the group G is primitive. For example, the group of symmetries of a square is imprimitive on the vertices: if they are numbered 1, 2, 3, 4 in cyclic order, then the partition {{1, 3}, {2, 4}} into opposite pairs is preserved by every group element. On the other hand, the full symmetric group on a set M is always primitive. Cayley's theorem Any group G can act on itself (the elements of the group being thought of as the set M) in many ways. In particular, there is a regular action given by (left) multiplication in the group. That is, f(g, x) = gx for all g and x in G. For each fixed g, the function fg(x) = gx is a bijection on G and therefore a permutation of the set of elements of G. Each element of G can be thought of as a permutation in this way and so G is isomorphic to a permutation group; this is the content of Cayley's theorem. For example, consider the group G1 acting on the set {1, 2, 3, 4} given above. Let the elements of this group be denoted by e, a, b and c = ab = ba. The action of G1 on itself described in Cayley's theorem gives the following permutation representation: fe ↦ (e)(a)(b)(c) fa ↦ (ea)(bc) fb ↦ (eb)(ac) fc ↦ (ec)(ab). Isomorphisms of permutation groups If G and H are two permutation groups on sets X and Y with actions f1 and f2 respectively, then we say that G and H are permutation isomorphic (or isomorphic as permutation groups) if there exists a bijective map and a group isomorphism such that λ(f1(g, x)) = f2(ψ(g), λ(x)) for all g in G and x in X. If this is equivalent to G and H being conjugate as subgroups of Sym(X). The special case where and ψ is the identity map gives rise to the concept of equivalent actions of a