Proving even permutations form a group under composition

[Braka]

Homework Statement

The question is:
Let A be a subset of S_n that contains all permutations alpha such that alpha can be written as a product of an even number of transpositions. Prove that A is a group with product of permutations.

I understand what I need to do to prove it, but I am not sure how to start it. Do I use:
Let alpha=(a1a2a3...an) and
beta=(b1b2b3...bn) and try to find closure,

alphabeta=(a1a2a3...an)(b1b2b3...bn)=(a1a2)(a1a3)...(a1an-1)(a1an)(b1b2)(b1b3)...(b1bn-1)(b1bn),

or am I going about it the wrong way?

Homework Equations

The Attempt at a Solution

 

[morphism]

What is a group? There are some axioms A must satisfy in order for it to be called one. For instance, the product of two things in A must also lie in A. Translating this to the situation at hand: if two permutations can be written as a product of an even number of transpositions, can their product be as well? Obviously yes! Now check the other axioms similarly.

 

[HallsofIvy]

Where did you use the fact that they can be written as an even number of transpositions?