Subgroup of a symmetric group Sn

leonamccauley

Homework Statement

Show that if G is a subgroup of a symmetric group Sn, then either every element of G is an even permutation or else exactly half the elements of G are even permutations.

The Attempt at a Solution

We have a hint for the problem. If all the elements of G are even, then there is nothing to prove. That is clear because even times even will yield and even number of permutations so there is closure under multiplication.
And if you have even transposition the inverse of it will be in the set as well so clearly if we have an even number of permutations its a subgroup.

If it is not we are to let e, o(1), 0(2),...o(k-1) be the even elements of G..

I am stuck so i looked at S3 a subgroup of S9. S3 has 6 elements.
(1 3 2) (1 2 3) and e are even
(1 2) (1 3) and (2 3) are odd
So exactly half like the theorem states but i am stuck.

Staff Emeritus
2) There exists an $$\sigma\in G_{odd}$$. Now try to show that $$G_{odd}=\{\sigma\tau~\vert~\tau \in G_{even}\}$$. Or in more highbrow terminology: Geven acts on Godd, and the orbit of an element in Godd is entire Godd.