cyclic permutation
Let $A=\{{a}_{0},{a}_{1},\mathrm{\dots},{a}_{n1}\}$ be a finite set^{} indexed by $i=0,\mathrm{\dots},n1$. A cyclic permutation^{} on $A$ is a permutation^{} $\pi $ on $A$ such that, for some integer $k$,
$$\pi ({a}_{i})={a}_{(i+k)\phantom{\rule{veryverythickmathspace}{0ex}}(modn)},$$ 
where $a\phantom{\rule{veryverythickmathspace}{0ex}}(modb):=a\lfloor a/b\rfloor b$, the remainder of $a$ when divided by $b$, and $\lfloor \cdot \rfloor $ is the floor function.
For example, if $A=\{1,2,\mathrm{\dots},m\}$ such that ${a}_{i}=i+1$. Then a cyclic permutation $\pi $ on $A$ has the form
$\pi (1)$  $=$  $r$  
$\pi (2)$  $=$  $r+1$  
$\mathrm{\vdots}$  
$\pi (mr+1)$  $=$  $m$  
$\pi (mr+2)$  $=$  $1$  
$\mathrm{\vdots}$  
$\pi (m)$  $=$  $r1.$ 
In the usual permutation notation, it looks like
$$\pi =\left(\begin{array}{ccccccc}\hfill 1\hfill & \hfill 2\hfill & \hfill \mathrm{\cdots}\hfill & \hfill mr+1\hfill & \hfill mr+2\hfill & \hfill \mathrm{\cdots}\hfill & \hfill m\hfill \\ \hfill r\hfill & \hfill r+1\hfill & \hfill \mathrm{\cdots}\hfill & \hfill m\hfill & \hfill 1\hfill & \hfill \mathrm{\cdots}\hfill & \hfill r1\hfill \end{array}\right)$$ 
Remark. For every finite set of cardinality $n$, there are $n$ cyclic permutations. Each nontrivial cyclic permutation has order $n$. Furthermore, if $n$ is a prime number^{}, the set of cyclic permutations forms a cyclic group^{}.
Cyclic permutations on words
Given a word $w={a}_{1}{a}_{2}\mathrm{\cdots}{a}_{n}$ on a set $\mathrm{\Sigma}$ (may or may not be finite), a cyclic conjugate of $w$ is a word $v$ derived from $w$ based on a cyclic permutation. In other words, $v=\pi ({a}_{1})\pi ({a}_{2})\mathrm{\cdots}\pi ({a}_{n})$ for some cyclic permutation $\pi $ on $\{{a}_{1},\mathrm{\dots},{a}_{n}\}$. Equivalently, $v$ and $w$ are cyclic conjugates of one another iff $w=st$ and $v=ts$ for some words $s,t$.
For example, the cyclic conjugates of the word $ababa$ over $\{a,b\}$ are
$$bab{a}^{2},ab{a}^{2}b,b{a}^{2}ba,{a}^{2}bab,\text{and itself}.$$ 
Strictly speaking, $\pi $ is a cyclic permutation on the multiset $A=\{{a}_{1},\mathrm{\dots},{a}_{n}\}$, which can be thought of as a cyclic permutation on the set ${A}^{\prime}=\{(1,{a}_{1}),\mathrm{\dots},(n,{a}_{n})\}$. Furthermore, $\pi $ can be extended to a function on ${A}^{*}$: for every word $w={a}_{\varphi (1)}\mathrm{\cdots}{a}_{\varphi (m)}$, $\pi (w):=\pi ({a}_{\varphi (1)})\mathrm{\cdots}\pi ({a}_{\varphi (m)})$, where $\varphi $ is a permutation on $A$.
Given any word $w={a}_{1}{a}_{2}\mathrm{\cdots}{a}_{n}$ on $\mathrm{\Sigma}$, two cyclic permutations ${\pi}_{1},{\pi}_{2}$ on $\{{a}_{1},\mathrm{\dots},{a}_{n}\}$ are said to be the same if ${\pi}_{1}(w)={\pi}_{2}(w)$. For example, with the word $abab$, then the cyclic permutation
$$\left(\begin{array}{cccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill \\ \hfill 3\hfill & \hfill 4\hfill & \hfill 1\hfill & \hfill 2\hfill \end{array}\right)$$ 
is the same as the identity^{} permutation. There is a onetoone correspondence between the set of all cyclic conjugates of $w$ and the set of all distinct cyclic permutations on $\{{a}_{0},{a}_{1},\mathrm{\dots},{a}_{n}\}$.
Remarks.

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In a group $G$, if two elements $u,v$ are cyclic conjugates of one another, then they are conjugates^{}: for if $u=st$ and $v=ts$, then $v=t(st){t}^{1}=tu{t}^{1}$.

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Cyclic permutations were used as a ciphering scheme by Julius Caesar. Given an alphabet with letters, say $a,b,c,\mathrm{\dots},x,y,z$, messages in letters are encoded so that each letter is shifted by three places. For example, the name
“Julius Caesar” becomes “Mxolxv Fdhvdu”.
A ciphering scheme based on cyclic permutations is therefore also known as a Caesar shift cipher.
Title  cyclic permutation 
Canonical name  CyclicPermutation 
Date of creation  20130322 17:33:54 
Last modified on  20130322 17:33:54 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  13 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 94B15 
Classification  msc 20B99 
Classification  msc 0300 
Classification  msc 05A05 
Classification  msc 11Z05 
Classification  msc 94A60 
Synonym  Caesar cipher 
Related topic  CyclicCode 
Related topic  SubgroupsWithCoprimeOrders 
Defines  Caesar shift cipher 
Defines  cyclic conjugate 