This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
View current documentation (Version 11.1)


gives the product of permutations a, b, c.
  • The product of permutations a, b, c is understood to be the permutation resulting from applying a, then b, then c.
  • The product of permutations is non-commutative.
Product of two permutations:
Multiplication of permutations is not commutative:
Product of two permutations:
Click for copyable input
Multiplication of permutations is not commutative:
Click for copyable input
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 these abbreviations:
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 permutations is equivalent to Part using permutation lists of adequate length obtained with PermutationList:
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:
New in 8