Proving the Subgroup Property of Even Permutations in Permutation Groups

[PhysicsUnderg]

Homework Statement

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.

 

[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.

 

[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?

 

[tiny-tim]

Hi PhysicsUnderg!

Is it really that simple?

Yes!

Sometimes, maths really is that simple! ​

… 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?

You ask "Is the identity an even permutation? What is the inverse of an even permutation?"