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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 every pair of elements of the set to an element 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.
Group theory has three main historical sources: number theory, the theory of algebraic equations, and geometry.The number-theoretic strand was begun by Leonhard Euler, and developed by Gauss's work on modular arithmetic and additive and multiplicative groups related to quadratic fields.
3. Between two groups, may mean that the first one is a proper subgroup of the second one. > (greater-than sign) 1. Strict inequality between two numbers; means and is read as "greater than". 2. Commonly used for denoting any strict order. 3. Between two groups, may mean that the second one is a proper subgroup of the first one. ≤ 1.
E.g. the Weyl group of a compact Lie group G with a torus T is defined as W(G,T) = N G (T)/C G (T), and especially if the torus is maximal (i.e. C G (T) = T) it is a central tool in the theory of Lie groups. C G (C G (S)) contains S, but C G (S) need not contain S. Containment occurs exactly when S is abelian. If H is a subgroup of G, then N G ...
A free group on a set S is a group where each element can be uniquely described as a finite length product of the form: . where the s i are elements of S, adjacent s i are distinct, and a i are non-zero integers (but n may be zero).
If E denotes the trivial group, G ≅ G × E ≅ E × G for any groups G. The order of a direct product G × H is the product of the orders of G and H: | G × H | = | G | | H |. This follows from the formula for the cardinality of the cartesian product of sets. The order of each element (g, h) is the least common multiple of the orders of g and ...
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If the quotient group G/Z(G) is cyclic, G is abelian (and hence G = Z(G), so G/Z(G) is trivial). The center of the Rubik's Cube group consists of two elements – the identity (i.e. the solved state) and the superflip. The center of the Pocket Cube group is trivial. The center of the Megaminx group has order 2, and the center of the Kilominx ...