# The collection of even-signed permutations is a 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.

Parents:

- Alternating group
The alternating group is the only normal subgroup of the symmetric group (on five or more generators).