A rotation permutation is, roughly speaking, a permutation in which, if we view the elements of a finite set as people standing in a circle, everybody shifts a given number of steps to the right, or everybody shifts a given number of steps to the left. It is an example of a cyclic permutation.
Recall that a permutation of a finite set can formally be defined to be a bijection .
Let be a finite set with elements. Fix an isomorphism, that is to say a bijection, . A rotation permutation of is a permutation of which belongs to the permutation group on generated by the permutation of such that the following diagram commutes, where is the permutation of given by for and by .
More explicitly, for each integer , there is a rotation permutation of given by for and by for . All rotation permutations of are of this form.
Let be a finite set with elements. Fix an isomorphism . A rotation permutation of is a permutation of such that the following diagram commutes, where is a rotation permutation of .
One sometimes also speaks of rotation of a word in some algebraic object.
Let be a word in some algebraic object, for example a free monoid or a free group. A rotation of is a word in the same algebraic object of the form where is a rotation permutation of .
Let be a set. Examples of a rotation of the word in (the free monoid on) are , , and .
Last revised on December 31, 2018 at 14:28:02. See the history of this page for a list of all contributions to it.