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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 ...
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.
If multiplying by an integer greater than 7, there is a simple process to get to a cyclic permutation of 142857. By adding the rightmost six digits (ones through hundred thousands) to the remaining digits and repeating this process until only six digits are left, it will result in a cyclic permutation of 142857: [citation needed] 142857 × 8 ...
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.
A cyclic number is an integer for which cyclic permutations of the digits are successive integer multiples of the number. The most widely known is the six-digit number 142857, whose first six integer multiples are 142857 × 1 = 142857 142857 × 2 = 285714 142857 × 3 = 428571 142857 × 4 = 571428 142857 × 5 = 714285 142857 × 6 = 857142
Along with the special linear group SL(2, 5) and the icosahedral group A 5 × S 2, S 5 is one of the three non-solvable groups of order 120, up to isomorphism. S 5 is the Galois group of the general quintic equation, and the fact that S 5 is not a solvable group translates into the non-existence of a general formula to solve quintic polynomials by
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.
A cyclic group is a group which is equal to one of its cyclic subgroups: G = g for some element g, called a generator of G. For a finite cyclic group G of order n we have G = {e, g, g 2, ... , g n−1}, where e is the identity element and g i = g j whenever i ≡ j (mod n); in particular g n = g 0 = e, and g −1 = g n−1.
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