Cyclically reduced word

In mathematics, cyclically reduced word is a concept of combinatorial group theory.

Let F(X) be a free group. Then a word w in F(X) is said to be cyclically reduced if and only if every cyclic permutation of the word is reduced.


  • Every cyclic shift and the inverse of a cyclically reduced word are cyclically reduced again.
  • Every word is conjugate to a cyclically reduced word. The cyclically reduced words are minimal-length representatives of the conjugacy classes in the free group. This representative is not uniquely determined, but it is unique up to cyclic shifts (since every cyclic shift is a conjugate element).


  • Solitar, Donald; Magnus, Wilhelm; Karrass, Abraham (1976), Combinatorial group theory: presentations of groups in terms of generators and relations, New York: Dover, pp. 33,188,212, ISBN 0-486-63281-4