Concept
Group theory
Related concepts (87)
Group (mathematics)
In mathematics, a group is a non-empty set with an operation that satisfies the following constraints: the operation is associative, has an identity element, and every element of the set has an inverse element. Many mathematical structures are groups endowed with other properties. For example, the integers with the addition operation is an infinite group, which is generated by a single element called 1 (these properties characterize the integers in a unique way).
Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a " matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra.
Normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup of the group is normal in if and only if for all and The usual notation for this relation is Normal subgroups are important because they (and only they) can be used to construct quotient groups of the given group.
Profinite group
In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups. The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups. Properties of the profinite group are generally speaking uniform properties of the system. For example, the profinite group is finitely generated (as a topological group) if and only if there exists such that every group in the system can be generated by elements.
Rubik's Cube
The Rubik's Cube is a 3-D combination puzzle originally invented in 1974 by Hungarian sculptor and professor of architecture Ernő Rubik. Originally called the Magic Cube, the puzzle was licensed by Rubik to be sold by Pentangle Puzzles in the UK in 1978, and then by Ideal Toy Corp in 1980 via businessman Tibor Laczi and Seven Towns founder Tom Kremer. The cube was released internationally in 1980 and became one of the most recognized icons in popular culture. It won the 1980 German Game of the Year special award for Best Puzzle.
Symmetry group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object. A frequent notation for the symmetry group of an object X is G = Sym(X). For an object in a metric space, its symmetries form a subgroup of the isometry group of the ambient space.
Lagrange's theorem (group theory)
In the mathematical field of group theory, Lagrange's theorem is a theorem that states that for any finite group G, the order (number of elements) of every subgroup of G divides the order of G. The theorem is named after Joseph-Louis Lagrange. The following variant states that for a subgroup of a finite group , not only is an integer, but its value is the index , defined as the number of left cosets of in . Lagrange's theorem This variant holds even if is infinite, provided that , , and are interpreted as cardinal numbers.
Word problem for groups
In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group G is the algorithmic problem of deciding whether two words in the generators represent the same element. More precisely, if A is a finite set of generators for G then the word problem is the membership problem for the formal language of all words in A and a formal set of inverses that map to the identity under the natural map from the free monoid with involution on A to the group G.
Finite group
In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. Important examples of finite groups include cyclic groups and permutation groups. The study of finite groups has been an integral part of group theory since it arose in the 19th century.
Symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group defined over a finite set of symbols consists of the permutations that can be performed on the symbols. Since there are ( factorial) such permutation operations, the order (number of elements) of the symmetric group is .