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The length of a cycle is the number of elements of its largest orbit. A cycle of length k is also called a k-cycle. The orbit of a 1-cycle is called a fixed point of the permutation, but as a permutation every 1-cycle is the identity permutation. [7] When cycle notation is used, the 1-cycles are often omitted when no confusion will result. [8]
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.
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".
The cycle graphs of dihedral groups consist of an n-element cycle and n 2-element cycles. The dark vertex in the cycle graphs below of various dihedral groups represents the identity element, and the other vertices are the other elements of the group. A cycle consists of successive powers of either of the elements connected to the identity element.
When a permutation is represented in cycle notation, the order of the cyclic subgroup that it generates is the least common multiple of the lengths of its cycles. For example, in S 5 , one cyclic subgroup of order 5 is generated by (13254), whereas the largest cyclic subgroups of S 5 are generated by elements like (123)(45) that have one cycle ...
The cycle index polynomial of a permutation group is the average of the cycle index monomials of its elements. The phrase cycle indicator is also sometimes used in place of cycle index. Knowing the cycle index polynomial of a permutation group, one can enumerate equivalence classes due to the group's action.
Each cycle has now become a Lyndon word, and they are arranged in lexicographic order, so dropping the parentheses yields the first de Bruijn sequence. For example, to construct the smallest B (2,4) de Bruijn sequence of length 2 4 = 16, repeat the alphabet (ab) 8 times yielding w =abababababababab .
A cycle graph illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups. A cycle graph for a cyclic group is simply a circular graph, where the group order is equal to the number of nodes. A single generator defines the group as a directional path on the graph, and the inverse ...