Permutation Groups

1. Jul 8, 2011

PhysicsUnderg

1. The problem statement, all variables and given/known data
Show that if G is any group of permutations, then the set of all even permutations in G forms a subgroup of G.

I am not sure where to start - I know there is a proposition that states this to be true, but I know that is not enough to prove this statement.

2. Jul 8, 2011

tiny-tim

Hi PhysicsUnderg!

Hint: if H is a subgroup of G, then for any a and b in H, the product ab must also be in H.

3. Jul 8, 2011

PhysicsUnderg

Is it really that simple? lol This is what I was thinking, but I wasn't sure how to connect the idea to permutations. Can I just say "a is an even permutation" and "b is an even permutation" thus "a*b is also an even permutation"? Because, if this is true and if I assume that a and b are elements of H, then ab is an element of H and is an even permutation, so G has a subgroup of even permutations. Also, for H to be a subgroup, the identity element must be contained in H, as well as an inverse. How do you connect this to permutations?

4. Jul 8, 2011

tiny-tim

Hi PhysicsUnderg!
Yes!!

Sometimes, maths really is that simple!
You ask "Is the identity an even permutation? What is the inverse of an even permutation?"