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 .