In mathematics, a group is a set equipped with a binary operation that combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study. Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; the concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s, who introduced the term of group for the symmetry group of the roots of an equation, now called a Galois group.
After contributions from other fields such as number theory and geometry, the group notion was generalized and established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists study the different ways in which a group can be expressed concretely, both from a point of view of representation theory and of computational group theory. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory. One of the most familiar groups is the set of integers Z which consists of the numbers... − 4, − 3, − − 1, 0, 1, 2, 3, 4... together with addition.
The following properties of integer addition serve as a model for the group axioms given in the definition below. For any two integers a and b, the sum a + b is an integer; that is, addition of integers always yields an integer. This property is known as closure under addition. For all integers a, b and c, + c = a +. Expressed in words, adding a to b first, adding the result to c gives the same final result as adding a to the sum of b and c, a property known as associativity. If a is any integer 0 + a = a + 0 = a. Zero is called the identity element of addition because adding it to any integer returns the same integer. For every integer a, there is an integer b such that a + b = b + a = 0; the integer b is called the inverse element of the integer a and is denoted −a. The integers, together with the operation +, form a mathematical object belonging to a broad class sharing similar structural aspects. To appropriately understand these structures as a collective, the following definition is developed.
A group is a set, G, together with an operation ⋅ that combines any two elements a and b to form another element, denoted a ⋅ b or ab. To qualify as a group, the set and operation, must satisfy four requirements known as the group axioms: Closure For all a, b in G, the result of the operation, a ⋅ b, is in G. Associativity For all a, b and c in G, ⋅ c = a ⋅. Identity element There exists an element e in G such that, for every element a in G, the equation e ⋅ a = a ⋅ e = a holds; such an element is unique, thus one speaks of the identity element. Inverse element For each a in G, there exists an element b in G denoted a−1, such that a ⋅ b = b ⋅ a = e, where e is the identity element; the result of the group operation may depend on the order of the operands. In other words, the result of combining element a with element b need not yield the same result as combining element b with element a; this equation always holds in the group of integers under addition, because a + b = b + a for any two integers.
Groups for which the commutativity equation a ⋅ b = b ⋅ a always holds are called abelian groups. The symmetry group described in the following section is an example of a group, not abelian; the identity element of a group G is written as 1 or 1G, a notation inherited from the multiplicative identity. If a group is abelian one may choose to denote the group operation by + and the identity element by 0; the identity element can be written as id. The set G is called the underlying set of the group; the group's underlying set G is used as a short name for the group. Along the same lines, shorthand expressions such as "a subset of the group G" or "an element of group G" are used when what is meant is "a subset of the underlying set G of the group" or "an element of the underlying set G of the group", it is clear from the context whether a symbol like G refers to a group or to an underlying set. An alternate definition is to expand the structure of a group to
The 1998 Stroud Council election took place on 7 May 1998 to elect members of Stroud District Council in Gloucestershire, England. One third of the council was up for the council stayed under no overall control. After the election, the composition of the council was Labour 26 Conservative 10 Liberal Democrat 9 Independent 6 Green 4 Before the election the Labour party ran the council, but the party had lost its one-seat overall majority after a vacancy in a Labour seat. 19 seats were contested in the election with Labour defending 10 seats, the Liberal Democrats 5, Green party 2 and independents 2. The results saw no party win a majority on the council after Labour lost 1 seat to the Conservatives and 1 seat to an independent
Gutierritos, was the name of the first telenovela produced in Mexico by Valentín Pimstein in 1958 for Telesistema Mexicano which aired Monday through Saturday at 6:30 pm on Channel 4. With an original script by Estella Calderón and directed and starred by Rafael Banquells besides Maria Teresa Rivas and Mauricio Garcés, Gutierritos is about a humble man abused by his family, co-workers. Ángel Gutiérrez is a kind and shy office employee who works hard for his family. His wife Rosa, who treats him badly and humiliates him all the time, despises him for being "so pathetic"; the children Julio Cesar and Lucrecia do not respect their father either after seeing their mother mistreating him and calling him "a pathetic, good for nothing". At the office everyone teases Angel including his boss Mr. Martinez who nicknames him "Gutierritos" which catches on among employees. Angel's only moral support at work is his friend Jorge. Mr. Martinez hires Elena an insecure but beautiful young woman. Angel secretly falls in love with Elena but it is in a book that Angel is writing that he expresses his feelings for her.
However, Angel shows the originals to his friend Jorge whom on steals them, claims the authorship, takes them to a publisher. Awed by "Jorge's talent" after reading the published book Elena falls in love with him. Angel tries to tell the truth but no one believes him while he suffers the loss of his friend, his book, the woman he loves. Jorge enjoys the fame, the success, Elena's love. Rafael Banquells - Ángel Gutiérrez "Gutierritos" María Teresa Rivas - Rosa Hernández Carlos Navarro as Juan Ortega Dina de Marco as Anita Manuel Lozano as Médina Vicky Aguirre as Lupita Mauricio Garcés as Jorge Contreras Patricia Morán as Elena Gerardo del Castillo as Señor Martínez Josefina Escobedo as Tía de Rosa Andrea López as Ágatha Elvira Quintana as Señora Gutiérrez Miguél Suárez as Señor Fernández María Eugenia Llamas as Lucrecia Luis de Alba as Julio Cesar