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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 satisfies the following constraints: the operation is associative, it has an identity element, and every element of the set has an inverse element. Many mathematical structures are groups
Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.
Given two groups (G, •) and (H, ·), an isomorphism between G and H is a bijective homomorphism from G to H, that is, a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. Two groups are isomorphic if there exists a group isomorphism mapping from one to the other. Isomorphic groups ...
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to represent group elements as invertible matrices so that the group operation can be represented by matrix ...
The beauty of mathematics is experienced when the physical reality of objects are represented by mathematical models. Group theory, developed in the early 1800s for the sole purpose of solving polynomial equations, became a fruitful way of categorizing elementary particles—the building blocks of matter.
In the mathematical area of group theory, Artin groups, also known as Artin–Tits groups or generalized braid groups, are a family of infinite discrete groups defined by simple presentations. They are closely related with Coxeter groups. Examples are free groups, free abelian groups, braid groups, and right-angled Artin–Tits groups, among others
In mathematics, a presentation is one method of specifying a group.A presentation of a group G comprises a set S of generators—so that every element of the group can be written as a product of powers of some of these generators—and a set R of relations among those generators.
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, namely a nontrivial normal subgroup and the corresponding quotient group .