# The collection of even-signed permutations is a group

This proves the well-definedness of one particular definition of the alternating group.

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.