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.

Properties

• 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).

References

• 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 

This algebra-related article is a stub. You can help Wikipedia by expanding it.v · Categories: Algebra stubs Combinatorial group theory Combinatorics on words Wikimedia Foundation. 2010. Игры ⚽ Нужна курсовая? Texas U-turn El Viaje Misterioso de Nuestro Jomer (The Mysterious Voyage of Homer) Look at other dictionaries: Word (group theory) — In group theory, a word is any written product of group elements and their inverses. For example, if x , y , and z are elements of a group G , then xy , z 1 xzz , and y 1 zxx 1 yz 1 are words in the set { x , y , z }. Words play an important role …   Wikipedia Small cancellation theory — In the mathematical subject of group theory, small cancellation theory studies groups given by group presentations satisfying small cancellation conditions, that is where defining relations have small overlaps with each other. It turns out that… …   Wikipedia List of mathematics articles (C) — NOTOC C C closed subgroup C minimal theory C normal subgroup C number C semiring C space C symmetry C* algebra C0 semigroup CA group Cabal (set theory) Cabibbo Kobayashi Maskawa matrix Cabinet projection Cable knot Cabri Geometry Cabtaxi number… …   Wikipedia Van Kampen diagram — In the mathematical area of geometric group theory, a van Kampen diagram is a planar diagram used to represent the fact that a particular word among the generators of a group given by a group presentation represents the identity element in that… …   Wikipedia Dehn function — In the mathematical subject of geometric group theory, a Dehn function, named after Max Dehn, is an optimal function associated to a finite group presentation which bounds the area of a relation in that group (that is a freely reduced word in the …   Wikipedia Out(Fn) — In mathematics, Out(Fn) is the outer automorphism group of a free group on n generators. These groups play an important role in geometric group theory. Contents 1 Outer space 1.1 Definition 1.2 Connection to length fun …   Wikipedia cryptology — cryptologist, n. cryptologic /krip tl oj ik/, cryptological, adj. /krip tol euh jee/, n. 1. cryptography. 2. the science and study of cryptanalysis and cryptography. [1635 45; < NL cryptologia. See CRYPTO , LOGY] * * * Introduction …   Universalium calendar — calendrical /keuh len dri keuhl/, calendric, calendarial /kal euhn dair ee euhl/, calendarian, calendaric, adj. /kal euhn deuhr/, n. 1. a table or register with the days of each month and week in a year: He marked the date on his calendar. 2. any …   Universalium Tibeto-Burman languages — Introduction       language group within the Sino Tibetan family (Sino Tibetan languages). At the end of the 20th century, Tibeto Burman languages were spoken by approximately 57 million people; countries that had more than 1 million Tibeto… …   Universalium Nielsen transformation — In mathematics, especially in the area of abstract algebra known as combinatorial group theory, Nielsen transformations, named after Jakob Nielsen, are certain automorphisms of a free group which are a non commutative analogue of row reduction… …   Wikipedia 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”