nLab rotation permutation

Idea

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.

Formal definition

Recall that a permutation of a finite set $X$ can formally be defined to be a bijection $X \rightarrow X$.

Definition

Let $X$ be a finite set with $n$ elements. Fix an isomorphism, that is to say a bijection, $i : X \rightarrow \{ 1, \ldots, n \}$. A rotation permutation of $X$ is a permutation of $X$ which belongs to the permutation group on $X$ generated by the permutation $p: X \rightarrow X$ of $X$ such that the following diagram commutes, where $\sigma$ is the permutation of $\{ 1, \ldots, n \}$ given by $i \mapsto i+1$ for $1 \leq i \leq n-1$ and by $n \mapsto 1$.

Remark

More explicitly, for each integer $0 \leq j \leq n$, there is a rotation permutation of $\{ 1, \ldots, n \}$ given by $i \mapsto j+i$ for $1 \leq i \leq n - j$ and by $i \mapsto i - (n - j)$ for $n - j + 1 \leq i \leq n$. All rotation permutations of $\{ 1, \ldots, n \}$ are of this form.

Remark

Let $X$ be a finite set with $n$ elements. Fix an isomorphism $i : X \rightarrow \{ 1, \ldots, n \}$. A rotation permutation of $X$ is a permutation $p: X \rightarrow X$ of $X$ such that the following diagram commutes, where $\sigma$ is a rotation permutation of $\{ 1, \ldots, n \}$.

Rotation of a word

One sometimes also speaks of rotation of a word in some algebraic object.

Definition

Let $w=a_{1} \cdots a_{n}$ be a word in some algebraic object, for example a free monoid or a free group. A rotation of $w$ is a word in the same algebraic object of the form $w=a_{\sigma(1)} \cdots a_{\sigma(n)}$ where $\sigma$ is a rotation permutation of $\{1, \ldots, n\}$.

Example

Let $X = \{a,b\}$ be a set. Examples of a rotation of the word $a^{2} b a b^{3} a$ in (the free monoid on) $X$ are $b a b^{3} a a^{2} = b a b^{3} a^{3}$, $a b^{3} a a^{2} b = a b^{3} a^{3} b$, and $a a^{2} b a b^{3} = a^{3} b a b^{3}$.