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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 ...
Not all permutations are cyclic permutations, but every permutation can be written as a product [5] of disjoint (having no common element) cycles in essentially one way. [6] As a permutation may have fixed points (elements that are unchanged by the permutation), these will be represented by cycles of length one.
Following the convention of omitting 1-cycles, one may interpret an individual cycle as a permutation which fixes all the elements not in the cycle (a cyclic permutation having only one cycle of length greater than 1). Then the list of disjoint cycles can be seen as the composition of these cyclic permutations.
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.
Any circulant is a matrix polynomial (namely, the associated polynomial) in the cyclic permutation matrix: = + + + + = (), where is given by the companion matrix = []. The set of n × n {\displaystyle n\times n} circulant matrices forms an n {\displaystyle n} - dimensional vector space with respect to addition and scalar multiplication.
The cycle lemma asserts that any sequence of A's and B's, where >, has precisely dominating cyclic permutations. To see this, just arrange the given sequence of p + q {\displaystyle p+q} A's and B's in a circle and repeatedly remove adjacent pairs AB until only p − q {\displaystyle p-q} A's remain.
Now from the fact that in a group if ab = e then ba = e, it follows that any cyclic permutation of the components of an element of X again gives an element of X. Therefore one can define an action of the cyclic group C p of order p on X by cyclic permutations of components, in other words in which a chosen generator of C p sends
Cyclic codes are a kind of block code with the property that the circular shift of a codeword will always yield another codeword. This motivates the following general definition: For a string s over an alphabet Σ , let shift ( s ) denote the set of circular shifts of s , and for a set L of strings, let shift ( L ) denote the set of all ...