cyclic permutation

Let A={a0,a1,,an-1} be a finite setMathworldPlanetmath indexed by i=0,,n-1. A cyclic permutationMathworldPlanetmath on A is a permutationMathworldPlanetmath π on A such that, for some integer k,


where a(modb):=a-a/bb, the remainder of a when divided by b, and is the floor function.

For example, if A={1,2,,m} such that ai=i+1. Then a cyclic permutation π on A has the form

π(1) = r
π(2) = r+1
π(m-r+1) = m
π(m-r+2) = 1
π(m) = r-1.

In the usual permutation notation, it looks like


Remark. For every finite set of cardinality n, there are n cyclic permutations. Each non-trivial cyclic permutation has order n. Furthermore, if n is a prime numberMathworldPlanetmath, the set of cyclic permutations forms a cyclic groupMathworldPlanetmath.

Cyclic permutations on words

Given a word w=a1a2an on a set Σ (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=π(a1)π(a2)π(an) for some cyclic permutation π on {a1,,an}. 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

baba2,aba2b,ba2ba,a2bab,and itself.

Strictly speaking, π is a cyclic permutation on the multiset A={a1,,an}, which can be thought of as a cyclic permutation on the set A={(1,a1),,(n,an)}. Furthermore, π can be extended to a function on A*: for every word w=aϕ(1)aϕ(m), π(w):=π(aϕ(1))π(aϕ(m)), where ϕ is a permutation on A.

Given any word w=a1a2an on Σ, two cyclic permutations π1,π2 on {a1,,an} are said to be the same if π1(w)=π2(w). For example, with the word abab, then the cyclic permutation


is the same as the identityPlanetmathPlanetmathPlanetmath permutation. There is a one-to-one correspondence between the set of all cyclic conjugates of w and the set of all distinct cyclic permutations on {a0,a1,,an}.


  • In a group G, if two elements u,v are cyclic conjugates of one another, then they are conjugatesPlanetmathPlanetmath: for if u=st and v=ts, then v=t(st)t-1=tut-1.

  • Cyclic permutations were used as a ciphering scheme by Julius Caesar. Given an alphabet with letters, say a,b,c,,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 2013-03-22 17:33:54
Last modified on 2013-03-22 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 03-00
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