Either all the permutations in H are even or

[rsa58]

Homework Statement

Show that for every subgroup H of S(n) [the symmetric group on n letters] for n>=2 either all the permutations in H are even or exactly half of them are even.

Homework Equations

The Attempt at a Solution

I didn't really know how to do this but i thought maybe since H is closed it has something to do with the fact that an odd permutation times an odd permutation produces an even permutation and an odd permutation times an even permutation produces an odd permutation.

 

[morphism]

Consider H$\cap$A(n) and HA(n). Use the fact that A(n) is an index 2 normal subgroup of S(n).

 

[Dick]

Even simpler, if H contains an odd permutation 'o', then H*o=H. But multiplying by o turns even permutations into odd, and vice versa.