# PermutationProduct   PermutationProduct[a,b,c]

gives the product of permutations a, b, c.

# Details • The product of permutations a, b, c is understood to be the permutation resulting from applying a, then b, then c.
• PermutationProduct[g1,g2,,gn] gives the left-to-right product of n permutations.
• The product of permutations is non-commutative.
• gives g.
• returns the identity permutation Cycles[{}].
• PermutationProduct[a,b] can be input as ab. The character is entered as p* or \[PermutationProduct].

# Examples

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## Basic Examples(1)

Product of two permutations:

Multiplication of permutations is not commutative:

## Scope(4)

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:

## Generalizations & Extensions(3)

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:

## Properties & Relations(5)

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:

## Possible Issues(1)

returns x, irrespectively of what x is: