PermutationOrder

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`PermutationOrder`

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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)Summary of the most common use cases

Find the order of a permutation:

In[1]:=1

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https://wolfram.com/xid/0dqvgen6a3m-llki6n

Out[1]=1

Scope (1)Survey of the scope of standard use cases

Find the order of a permutation with any support:

In[1]:=1

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https://wolfram.com/xid/0dqvgen6a3m-xs22e6

Out[1]=1

Applications (1)Sample problems that can be solved with this function

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:

In[1]:=1

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https://wolfram.com/xid/0dqvgen6a3m-k2y98l

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0dqvgen6a3m-oedmbc

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0dqvgen6a3m-lylokk

Out[3]=3

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

In[4]:=4

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https://wolfram.com/xid/0dqvgen6a3m-24oh0r

Properties & Relations (6)Properties of the function, and connections to other functions

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

In[1]:=1

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https://wolfram.com/xid/0dqvgen6a3m-onnk8l

Out[1]=1

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

In[1]:=1

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https://wolfram.com/xid/0dqvgen6a3m-0buqso

In[2]:=2

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https://wolfram.com/xid/0dqvgen6a3m-9o2zs

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0dqvgen6a3m-p7pxma

Out[3]=3

In[4]:=4

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https://wolfram.com/xid/0dqvgen6a3m-myadrr

Out[4]=4

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

In[1]:=1

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https://wolfram.com/xid/0dqvgen6a3m-0syhmk

In[2]:=2

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https://wolfram.com/xid/0dqvgen6a3m-6odt99

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0dqvgen6a3m-valdhg

Out[3]=3

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:

In[1]:=1

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https://wolfram.com/xid/0dqvgen6a3m-vhjba6

Out[1]=1

There is no permutation with order 6:

In[2]:=2

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https://wolfram.com/xid/0dqvgen6a3m-oeqciw

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0dqvgen6a3m-4gz91l

Out[3]=3

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:

In[1]:=1