# 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).