# PermutationOrder

PermutationOrder[perm]

gives the order of permutation perm.

# Details

• The order of a permutation perm is the smallest positive integer m so that the product of perm with itself m times yields the identity permutation.
• The only permutation with order 1 is the identity permutation.

# Examples

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## Basic Examples(1)

Find the order of a permutation:

## Scope(1)

Find the order of a permutation with any support:

## Applications(1)

Group elements with order 2 are called involutions. If all elements of a group (except the identity) have order 2, then the group is Abelian (the opposite implication does not hold). This group is Abelian:

The group is Abelian because its multiplication table is symmetric. The involution character of all group elements is expressed by the diagonal of 1s:

## Properties & Relations(6)

The order of the identity permutation is defined to be 1:

The order of a permutation can be computed as the least common multiple of the lengths of its cycles:

The order of a permutation equals the order of the cyclic group generated by that permutation:

By Lagrange's theorem, the order of each element of a finite group divides the order of the group. However, not all divisors of the order of a group correspond to the order of some element in the group. Take the alternating group of degree 4, which has order 12, and hence divisors 6, 3, 2:

There is no permutation with order 6:

Cauchy's theorem states that for every prime divisor of the order of a group, there is an element in the group with that order. Take the alternating group of degree 7:

These are the factorization of the order and the orders present:

These are examples of permutations of the 4 prime orders:

Numbers of permutations in with different orders:

Generating function of order 6, for all symmetric groups:

Number of permutations in with order 6:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_permutationorder, author="Wolfram Research", title="{PermutationOrder}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/PermutationOrder.html}", note=[Accessed: 16-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_permutationorder, organization={Wolfram Research}, title={PermutationOrder}, year={2010}, url={https://reference.wolfram.com/language/ref/PermutationOrder.html}, note=[Accessed: 16-September-2024 ]}