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∩An.
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.