cyclically reduced

Let M(X) be a free monoid with involution -1 on X. A word wM(X) is said to be cyclically reduced if every cyclic conjugate of it is reduced. In other words, w is cyclically reduced iff w is a reduced word and that if w=uvu-1 for some words u and v, then w=v.

For example, if X={a,b,c}, then words such as

c-1bc2a  and  abac2ba2

are cyclically reduced, where as words

a2bca-1  and  cb2b3c

are not, the former is reduced, but of the form a(abc)a-1, while the later is not even a reduced word.

Remarks. The concept of cyclically reduced words carries over to words in groups. We consider words in a group G.

  • If a word is cyclically reduced, so is its inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath and all of its cyclic conjugates.

  • A word v is a cyclic reduction of a word w if w=uvu-1 for some word u, and v is cyclically reduced. Clearly, every word and its cyclic reduction are conjugates of each other. Furthermore, any word has a unique cyclic reduction.

  • Every group G has a presentationMathworldPlanetmathPlanetmathPlanetmath S|R such that

    1. (a)

      R is cyclically reduced (meaning every element of R is cyclically reduced),

    2. (b)

      closed underPlanetmathPlanetmath inverses (meaning if uR, then u-1R), and

    3. (c)

      closed under cyclic conjugation (meaning any cyclic conjugate of an element in R is in R).

    Furthermore, if G is finitely presented, R above can be chosen to be finite.


    Every group G has a presentation S|R. There is an isomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath from F(S)/N(R) to G, where F(S) is the free groupMathworldPlanetmath freely generated by S, and N(R) is the normalizerMathworldPlanetmath of the subset RF(S) in F(S). Let R′′ be the set of all cyclic reductions of words in R. Then N(R′′)=N(R), since any word not cyclically reduced in R is conjugate to its cyclic reduction in R′′, and hence in N(R′′). Next, for each uR′′, toss in its inverse and all of its cyclic conjugates. The resulting set R is still cyclically reduced. Furthermore, R satisfies the remaining conditions above. Finally, N(R)=N(R′′), as any cyclic conjugate v of a word w is clearly a conjugate of w. Therefore, G has presentation S|R.

    If G is finitely presented, then S and R above can be chosen to be finite setsMathworldPlanetmath. Therefore, R′′ and R are both finite. R is finite because the number of cyclic conjugates of a word is at most the length of the word, and hence finite. ∎

Title cyclically reduced
Canonical name CyclicallyReduced
Date of creation 2013-03-22 17:34:04
Last modified on 2013-03-22 17:34:04
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Definition
Classification msc 20F10
Classification msc 20F05
Classification msc 20F06
Defines cyclic reduction
Defines cyclic conjugation