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gives the order of permutation perm.
  • 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.
Find the order of a permutation:
Find the order of a permutation:
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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:
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