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The manipulations of the Rubik's Cube form the Rubik's Cube group.. In mathematics, a group is a set with an operation that associates an element of the set to every pair of elements of the set (as does every binary operation) and satisfies the following constraints: the operation is associative, it has an identity element, and every element of the set has an inverse element.
The map m is the group operation, the map e (whose domain is a singleton) picks out the identity element u of G, and the map inv assigns to every group element its inverse. e G : G → G is the map that sends every element of G to the identity element. A topological group is a group object in the category of topological spaces with continuous ...
In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S . Many sets of transformations form a group under function composition ; for example, the rotations around a point in the plane.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra.
The zero objects in Grp are the trivial groups (consisting of just an identity element). Every morphism f : G → H in Grp has a category-theoretic kernel (given by the ordinary kernel of algebra ker f = { x in G | f ( x ) = e }), and also a category-theoretic cokernel (given by the factor group of H by the normal closure of f ( G ) in H ).
If G is any group, and S is a generating subset of G, then every element of G is also of the above form; but in general, these products will not uniquely describe an element of G. For example, the dihedral group D 8 of order sixteen can be generated by a rotation, r, of order 8; and a flip, f, of order 2; and certainly any element of D 8 is a ...
A comparatively recent trend in the theory of finite groups exploits their connections with compact topological groups (profinite groups): for example, a single p-adic analytic group G has a family of quotients which are finite p-groups of various orders, and properties of G translate into the properties of its finite quotients.
The representation theory of groups divides into subtheories depending on the kind of group being represented. The various theories are quite different in detail, though some basic definitions and concepts are similar. The most important divisions are: Finite groups — Group representations are a very important tool in the study of finite groups.