Finite Groups, Simple Groups Consist of Even Permutations

Simple Groups Consist of Even Permutations

Let G be a permutation group with at least one element x that is an odd permutation. Let H = G∩A_n. Thus H is the elements of G that are even permutations. Multiply H by x to get a map from H onto the elements of G that are odd permutations. This is a 1-1 map, hence there are just as many elements in H as there are outside of H. Since H has index 2 it is normal in G. Seen another way, H is the kernel of the parity homomorphism.

If G is simple then H is trivial. This means G is a single involution, an odd number of disjoint transpositions. Other than S₂, a permutation group that is simple contains only even permutations.