Find the order of a permutation:

Find the order of a permutation with any support:

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:

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: