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This has the intuitive meaning that the images of x and y are supposed to be equal in the quotient group. Thus, for example, r n in the list of relators is equivalent with =. [1] For a finite group G, it is possible to build a presentation of G from the group multiplication table, as follows.
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.
A cyclic group is a group which is equal to one of its cyclic subgroups: G = g for some element g, called a generator of G. For a finite cyclic group G of order n we have G = {e, g, g 2, ... , g n−1}, where e is the identity element and g i = g j whenever i ≡ j (mod n); in particular g n = g 0 = e, and g −1 = g n−1.
A subgroup N of a group G is a normal subgroup of G if and only if for all elements g of G the corresponding left and right cosets are equal, that is, gN = Ng. This is the case for the subgroup H in the first example above. Furthermore, the cosets of N in G form a group called the quotient group or factor group G / N.
In set theory, any two sets are defined to be equal if they have all the same members. This is called the Axiom of extensionality. Usually set theory is defined within logic, and therefore uses the equality described above, however, if a logic system does not have equality, it is possible to define equality within set theory.
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 ...
Some authors define the commutator as [g, h] = ghg −1 h −1 instead. The commutator of two elements g and h is equal to the group's identity if and only if g and h commutate, that is, if and only if gh = hg. commutator subgroup The commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the ...
A normal subgroup of a normal subgroup of a group need not be normal in the group. That is, normality is not a transitive relation. The smallest group exhibiting this phenomenon is the dihedral group of order 8. [15] However, a characteristic subgroup of a normal subgroup is normal. [16] A group in which normality is transitive is called a T ...