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The size n of the orbit is called the length of the corresponding cycle; when n = 1, the single element in the orbit is called a fixed point of the permutation. A permutation is determined by giving an expression for each of its cycles, and one notation for permutations consist of writing such expressions one after another in some order.
A cyclic permutation consisting of a single 8-cycle. There is not widespread consensus about the precise definition of a cyclic permutation. Some authors define a permutation σ of a set X to be cyclic if "successive application would take each object of the permuted set successively through the positions of all the other objects", [1] or, equivalently, if its representation in cycle notation ...
Cycle notation describes the effect of repeatedly applying the permutation on the elements of the set S, with an orbit being called a cycle. The permutation is written as a list of cycles; since distinct cycles involve disjoint sets of elements, this is referred to as "decomposition into disjoint cycles".
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The cycle index of a permutation group G is the average of the cycle index monomials of all the permutations g in G. More formally, let G be a permutation group of order m and degree n . Every permutation g in G has a unique decomposition into disjoint cycles, say c 1 c 2 c 3 ... .
A sequence of numbers which after repeated application of the permutation returns to the first number is called a cycle of the permutation. Every permutation can be decomposed into disjoint cycles, that is, cycles which have no common elements. The permutation of the first example above can be written in cycle notation as
Equivalently, g(n) is the largest least common multiple (lcm) of any partition of n, or the maximum number of times a permutation of n elements can be recursively applied to itself before it returns to its starting sequence. For instance, 5 = 2 + 3 and lcm(2,3) = 6. No other partition of 5 yields a bigger lcm, so g(5) = 6.
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