Permutation group

The group of all permutations of the set .

Unique factorisation

Each has a factorisation in where are disjoint cycles.
This factorisation is unique up to permutations of .
Cycle Type

Unique sign

Each can be written as a product of transpositions
(for example, by decomposing each cycle into a product of transpositions).
Then there is a Homomorphism
where if can be written with an even number of transpositions,
while if it can’t.

Proposition

This is well defined.

It induces an Alternating group .