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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 index; Cycle notation; Cycles and fixed points; Cyclic order; Direct sum of permutations; Enumerations of specific permutation classes; Factorial. Falling factorial; Permutation matrix. Generalized permutation matrix; Inversion (discrete mathematics) Major index; Ménage problem; Permutation graph; Permutation pattern; Permutation ...
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".
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
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 ... .
There are a few equivalent ways to state this definition. A cyclic order on X is the same as a permutation that makes all of X into a single cycle, which is a special type of permutation - a circular permutation. Alternatively, a cycle with n elements is also a Z n-torsor: a set with a free transitive action by a finite cyclic group. [1]
The black arrows indicate disjoint cycles and correspond to cycle notation. Green circle is an odd permutation, white is an even permutation and black is the identity. These are the positions of the six matrices Some matrices are not arranged symmetrically to the main diagonal – thus the symmetric group is not abelian.