PermutationProduct

✖
PermutationProduct
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.
- PermutationProduct[g] gives g.
- PermutationProduct[] returns the identity permutation Cycles[{}].
- PermutationProduct[a,b] can be input as ab. The character is entered as
p*
or \[PermutationProduct].
Examples
open allclose allBasic Examples (1)Summary of the most common use cases
Scope (4)Survey of the scope of standard use cases
PermutationProduct works with any number of permutations, of any degree:

https://wolfram.com/xid/0n4myrttwwa-y8suff

Product of a single permutation:

https://wolfram.com/xid/0n4myrttwwa-sc77he

Multiplication with the identity permutation:

https://wolfram.com/xid/0n4myrttwwa-5nxi4y

This gives the identity permutation:

https://wolfram.com/xid/0n4myrttwwa-777kou

Generalizations & Extensions (3)Generalized and extended use cases
PermutationProduct performs some simplifications with symbolic arguments:

https://wolfram.com/xid/0n4myrttwwa-e2x5tu


https://wolfram.com/xid/0n4myrttwwa-guvjrz


https://wolfram.com/xid/0n4myrttwwa-dc4j9l

Perform intermediate products:

https://wolfram.com/xid/0n4myrttwwa-g7bj0j

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

https://wolfram.com/xid/0n4myrttwwa-70senk

https://wolfram.com/xid/0n4myrttwwa-zhjb5l

https://wolfram.com/xid/0n4myrttwwa-yy0vcx
Two permutations commute if and only if their commutator is the identity:

https://wolfram.com/xid/0n4myrttwwa-9cm0jb

Commutation can be recursively generalized to more arguments:

https://wolfram.com/xid/0n4myrttwwa-j6cikf
Check some well-known commutation relations:

https://wolfram.com/xid/0n4myrttwwa-jkjzca


https://wolfram.com/xid/0n4myrttwwa-uz7l87

Properties & Relations (5)Properties of the function, and connections to other functions
Multiplication with the inverse permutation returns the identity:

https://wolfram.com/xid/0n4myrttwwa-5l4ivq


https://wolfram.com/xid/0n4myrttwwa-mwtwkj


https://wolfram.com/xid/0n4myrttwwa-qi3ynr

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

https://wolfram.com/xid/0n4myrttwwa-id6cdc

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

https://wolfram.com/xid/0n4myrttwwa-ojdydz



https://wolfram.com/xid/0n4myrttwwa-dceeil


https://wolfram.com/xid/0n4myrttwwa-pajj01

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

https://wolfram.com/xid/0n4myrttwwa-q7v32d

https://wolfram.com/xid/0n4myrttwwa-gledns

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

https://wolfram.com/xid/0n4myrttwwa-mpm0i0


https://wolfram.com/xid/0n4myrttwwa-lc0d0t


https://wolfram.com/xid/0n4myrttwwa-h5e8h8


https://wolfram.com/xid/0n4myrttwwa-0vyxqm

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

https://wolfram.com/xid/0n4myrttwwa-dlikdc


https://wolfram.com/xid/0n4myrttwwa-jvkdcf


https://wolfram.com/xid/0n4myrttwwa-4cs54o

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

https://wolfram.com/xid/0n4myrttwwa-qemkdg

Possible Issues (1)Common pitfalls and unexpected behavior
PermutationProduct[x] returns x, irrespectively of what x is:

https://wolfram.com/xid/0n4myrttwwa-ihdemj

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.
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.
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
Wolfram Language. (2010). PermutationProduct. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PermutationProduct.html
BibTeX
@misc{reference.wolfram_2025_permutationproduct, author="Wolfram Research", title="{PermutationProduct}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/PermutationProduct.html}", note=[Accessed: 29-March-2025
]}
BibLaTeX
@online{reference.wolfram_2025_permutationproduct, organization={Wolfram Research}, title={PermutationProduct}, year={2010}, url={https://reference.wolfram.com/language/ref/PermutationProduct.html}, note=[Accessed: 29-March-2025
]}