# 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:

Wolfram Research (2010), PermutationProduct, Wolfram Language function, https://reference.wolfram.com/language/ref/PermutationProduct.html.

#### Text

Wolfram Research (2010), PermutationProduct, Wolfram Language function, https://reference.wolfram.com/language/ref/PermutationProduct.html.

#### CMS

Wolfram Language. 2010. "PermutationProduct." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PermutationProduct.html.

#### APA

Wolfram Language. (2010). PermutationProduct. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PermutationProduct.html

#### BibTeX

@misc{reference.wolfram_2024_permutationproduct, author="Wolfram Research", title="{PermutationProduct}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/PermutationProduct.html}", note=[Accessed: 18-June-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_permutationproduct, organization={Wolfram Research}, title={PermutationProduct}, year={2010}, url={https://reference.wolfram.com/language/ref/PermutationProduct.html}, note=[Accessed: 18-June-2024 ]}