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 xzz and y zxx yz are words in the set { x , y , z }. Two different words may evaluate to the same value in G , or even in every group. Words play an important role in the theory of free groups and presentations , and are central objects of study in combinatorial group theory .
29-458: [REDACTED] Look up Factor or factor in Wiktionary, the free dictionary. Factor (Latin, 'who/which acts') may refer to: Commerce [ edit ] Factor (agent) , a person who acts for, notably a mercantile and colonial agent Factor (Scotland) , a person or firm managing a Scottish estate Factors of production , such
58-445: A factor is a resource used in the production of goods and services Factor, a brand of HelloFresh meal-kit company Science and technology [ edit ] Biology [ edit ] Coagulation factors , substances essential for blood coagulation Environmental factor , any abiotic or biotic factor that affects life Enzyme , proteins that catalyze chemical reactions Factor B , and factor D , peptides involved in
87-504: A member or component of a chord FACTOR , the Foundation to Assist Canadian Talent on Records The Factor , an April 2017 TV show on Fox News Channel See also [ edit ] [REDACTED] Search for "factor" on Misplaced Pages. All pages with titles containing factor All pages with titles beginning with Factor Co-factor (disambiguation) Factoring (disambiguation) Topics referred to by
116-451: A member or component of a chord FACTOR , the Foundation to Assist Canadian Talent on Records The Factor , an April 2017 TV show on Fox News Channel See also [ edit ] [REDACTED] Search for "factor" on Misplaced Pages. All pages with titles containing factor All pages with titles beginning with Factor Co-factor (disambiguation) Factoring (disambiguation) Topics referred to by
145-482: A multiplication Divisor , an integer which evenly divides a number without leaving a remainder Factorization , the decomposition of an object into a product of other objects Integer factorization , the process of breaking down a composite number into smaller non-trivial divisors A coefficient , a multiplicative factor in an expression, usually a number The act of forming a factor group or quotient ring in abstract algebra A von Neumann algebra , with
174-482: A multiplication Divisor , an integer which evenly divides a number without leaving a remainder Factorization , the decomposition of an object into a product of other objects Integer factorization , the process of breaking down a composite number into smaller non-trivial divisors A coefficient , a multiplicative factor in an expression, usually a number The act of forming a factor group or quotient ring in abstract algebra A von Neumann algebra , with
203-522: A person or firm managing a Scottish estate Factors of production , such a factor is a resource used in the production of goods and services Factor, a brand of HelloFresh meal-kit company Science and technology [ edit ] Biology [ edit ] Coagulation factors , substances essential for blood coagulation Environmental factor , any abiotic or biotic factor that affects life Enzyme , proteins that catalyze chemical reactions Factor B , and factor D , peptides involved in
232-463: A set of correlated variables. People [ edit ] Factor Chandelier , Canadian hip hop artist John Factor (1892–1984), British-American Prohibition-era gangster Max Factor Sr. (1872–1938), Polish-American businessman and cosmetician Max Factor Jr. (1904–1996), son of the above, born Francis Factor Places [ edit ] Factor, Arecibo, Puerto Rico , a barrio Other uses [ edit ] Factor (chord) ,
261-463: A set of correlated variables. People [ edit ] Factor Chandelier , Canadian hip hop artist John Factor (1892–1984), British-American Prohibition-era gangster Max Factor Sr. (1872–1938), Polish-American businessman and cosmetician Max Factor Jr. (1904–1996), son of the above, born Francis Factor Places [ edit ] Factor, Arecibo, Puerto Rico , a barrio Other uses [ edit ] Factor (chord) ,
290-404: A string Authentication factor , a piece of information used to verify a person's identity for security purposes Decomposition (computer science) , also known as factoring, the organization of computer code Enumerated type : a data type consisting of a set of named values, called factor in the R programming language Other uses in science and technology [ edit ] Factor, in
319-404: A string Authentication factor , a piece of information used to verify a person's identity for security purposes Decomposition (computer science) , also known as factoring, the organization of computer code Enumerated type : a data type consisting of a set of named values, called factor in the R programming language Other uses in science and technology [ edit ] Factor, in
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#1732845409158348-457: A trivial center Factor (graph theory) , a spanning sub graph Any finite contiguous sub-sequence of a word in combinatorics or of a word in group theory . Statistics [ edit ] An independent categorical variable . In experimental design , the factor is a category of treatments controlled by the experimenter. In factor analysis , the factors are unobserved underlying hidden variables that explain variability in
377-457: A trivial center Factor (graph theory) , a spanning sub graph Any finite contiguous sub-sequence of a word in combinatorics or of a word in group theory . Statistics [ edit ] An independent categorical variable . In experimental design , the factor is a category of treatments controlled by the experimenter. In factor analysis , the factors are unobserved underlying hidden variables that explain variability in
406-519: Is a generating set for a group G , a relation is a pair of words in S that represent the same element of G . These are usually written as equations, e.g. x − 1 y x = y 2 . {\displaystyle x^{-1}yx=y^{2}.\,} A set R {\displaystyle {\mathcal {R}}} of relations defines G if every relation in G follows logically from those in R {\displaystyle {\mathcal {R}}} using
435-407: Is a word that contains no redundant pairs. Any word can be simplified to a reduced word by performing a sequence of reductions: The result does not depend on the order in which the reductions are performed. A word is cyclically reduced if and only if every cyclic permutation of the word is reduced. The product of two words is obtained by concatenation: Even if the two words are reduced,
464-445: Is different from Wikidata All article disambiguation pages All disambiguation pages Factor [REDACTED] Look up Factor or factor in Wiktionary, the free dictionary. Factor (Latin, 'who/which acts') may refer to: Commerce [ edit ] Factor (agent) , a person who acts for, notably a mercantile and colonial agent Factor (Scotland) ,
493-438: Is different from Wikidata All article disambiguation pages All disambiguation pages Word (group theory) Let G be a group, and let S be a subset of G . A word in S is any expression of the form where s 1 ,..., s n are elements of S , called generators , and each ε i is ±1. The number n is known as the length of the word. Each word in S represents an element of G , namely
522-481: Is not a generating set for G , the set of elements represented by words in S is a subgroup of G , known as the subgroup of G generated by S and usually denoted ⟨ S ⟩ {\displaystyle \langle S\rangle } . It is the smallest subgroup of G that contains the elements of S . A normal form for a group G with generating set S is a choice of one reduced word in S for each element of G . For example: If S
551-473: The axioms for a group . A presentation for G is a pair ⟨ S ∣ R ⟩ {\displaystyle \langle S\mid {\mathcal {R}}\rangle } , where S is a generating set for G and R {\displaystyle {\mathcal {R}}} is a defining set of relations. For example, the Klein four-group can be defined by the presentation Here 1 denotes
580-405: The design of experiments , a phenomenon presumed to affect an experiment Human factors , a profession that focuses on how people interact with products, tools, or procedures Sun protection factor , a unit describing reduction in transmitted ultraviolet light Mathematics [ edit ] General mathematics [ edit ] Factor (arithmetic) , either of two numbers involved in
609-405: The design of experiments , a phenomenon presumed to affect an experiment Human factors , a profession that focuses on how people interact with products, tools, or procedures Sun protection factor , a unit describing reduction in transmitted ultraviolet light Mathematics [ edit ] General mathematics [ edit ] Factor (arithmetic) , either of two numbers involved in
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#1732845409158638-400: The above word would be written as follows: Any word in which a generator appears next to its own inverse ( xx or x x ) can be simplified by omitting the redundant pair: This operation is known as reduction , and it does not change the group element represented by the word. Reductions can be thought of as relations (defined below ) that follow from the group axioms . A reduced word
667-434: The alternate pathway of immune system complement activation Transcription factor , a protein that binds to specific DNA sequences Computer science and information technology [ edit ] Factor (programming language) , a concatenative stack-oriented programming language Factor (Unix) , a utility for factoring an integer into its prime factors Factor, a substring , a subsequence of consecutive symbols in
696-434: The alternate pathway of immune system complement activation Transcription factor , a protein that binds to specific DNA sequences Computer science and information technology [ edit ] Factor (programming language) , a concatenative stack-oriented programming language Factor (Unix) , a utility for factoring an integer into its prime factors Factor, a substring , a subsequence of consecutive symbols in
725-409: The empty word, which represents the identity element. If S is any set, the free group over S is the group with presentation ⟨ S ∣ ⟩ {\displaystyle \langle S\mid \;\rangle } . That is, the free group over S is the group generated by the elements of S , with no extra relations. Every element of the free group can be written uniquely as
754-421: The product may not be. The inverse of a word is obtained by inverting each generator, and reversing the order of the elements: The product of a word with its inverse can be reduced to the empty word: You can move a generator from the beginning to the end of a word by conjugation : A subset S of a group G is called a generating set if every element of G can be represented by a word in S . When S
783-526: The product of the expression. By convention, the unique identity element can be represented by the empty word , which is the unique word of length zero. When writing words, it is common to use exponential notation as an abbreviation. For example, the word could be written as This latter expression is not a word itself—it is simply a shorter notation for the original. When dealing with long words, it can be helpful to use an overline to denote inverses of elements of S . Using overline notation,
812-463: The same term [REDACTED] This disambiguation page lists articles associated with the title Factor . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Factor&oldid=1253769567 " Categories : Disambiguation pages Disambiguation pages with surname-holder lists Hidden categories: Short description
841-463: The same term [REDACTED] This disambiguation page lists articles associated with the title Factor . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Factor&oldid=1253769567 " Categories : Disambiguation pages Disambiguation pages with surname-holder lists Hidden categories: Short description
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