Permutation Group - Isomorphisms

Isomorphisms

If G and H are two permutation groups on the same set X, then we say that G and H are isomorphic as permutation groups if there exists a bijective map f : X → X such that r ↦ f −1 o r o f defines a bijective map between G and H; in other words, if for each element g in G, there is a unique h_g in H such that for all x in X, (g o f)(x) = (f o h_g)(x). This is equivalent to G and H being conjugate as subgroups of Sym(X). In this case, G and H are also isomorphic as groups.

Notice that different permutation groups may well be isomorphic as abstract groups, but not as permutation groups. For instance, the permutation group on {1,2,3,4} described above is isomorphic as a group (but not as a permutation group) to {(1)(2)(3)(4), (12)(34), (13)(24), (14)(23)}. Both are isomorphic as groups to the Klein group V₄.