The collection of elements of the Symmetric group $~$S_n$~$ which are made by multiplying together an even number of permutations forms a subgroup of $~$S_n$~$.

This proves that the Alternating group $~$A_n$~$ is well-defined, if it is given as "the subgroup of $~$S_n$~$ containing precisely that which is made by multiplying together an even number of transpositions".

# Proof

Firstly we must check that "I can only be made by multiplying together an even number of transpositions" is a well-defined notion; this is in fact true.

We must check the group axioms.

- Identity: the identity is simply the product of no transpositions, and $~$0$~$ is even.
- Associativity is inherited from $~$S_n$~$.
- Closure: if we multiply together an even number of transpositions, and then a further even number of transpositions, we obtain an even number of transpositions.
- Inverses: if $~$\sigma$~$ is made of an even number of transpositions, say $~$\tau_1 \tau_2 \dots \tau_m$~$, then its inverse is $~$\tau_m \tau_{m-1} \dots \tau_1$~$, since a transposition is its own inverse.