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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.
In abstract algebra, group theory studies the algebraic structures known as groups. 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 ...
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 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 ...
In mathematics, the order of a finite group is the number of its elements. ... By definition, the order of the identity, e, is one, since e 1 = e.
In mathematics, specifically group theory, the index of a subgroup H in a group G is the number of left cosets of H in G, ... since Ha ⊂ H by the definition of A.
The group consisting of all permutations of a set M is the symmetric group of M. p-group If p is a prime number, then a p-group is one in which the order of every element is a power of p. A finite group is a p-group if and only if the order of the group is a power of p. p-subgroup A subgroup that is also a p-group.
The rank of a symmetry group is closely related to the complexity of the object (a molecule, a crystal structure) being under the action of the group. If G is a crystallographic point group, then rank(G) is up to 3. [9] If G is a wallpaper group, then rank(G) = 2 to 4. The only wallpaper-group type of rank 4 is p2mm. [10]