Product of two permutations:

Multiplication of permutations is not commutative:

PermutationProduct works with any number of permutations, of any degree:

Product of a single permutation:

Multiplication with the identity permutation:

This gives the identity permutation:

PermutationProduct performs some simplifications with symbolic arguments:

Perform intermediate products:

From the product and inversion in a group, it is possible to define commutation and conjugation as follows. Use this abbreviation:

Define:

Two permutations commute if and only if their commutator is the identity:

Commutation can be recursively generalized to more arguments:

Check some well-known commutation relations:

Multiplication with the inverse permutation returns the identity:

Any cycle of length is equivalent to a product of transpositions (cycles of length 2) all having the same first point:

Multiplication of permutation lists is equivalent to Part but reversing the order:

Repeated multiplication of a single permutation can be computed with PermutationPower:

The product of all elements of a group depends on the order in which the product is computed:

For an Abelian group, the result is unique. In particular, for a cyclic group the result is very simple:

The result is simply this power of the generator of the cyclic group:

PermutationProduct[x] returns x, irrespectively of what x is: