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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 mathematics this group is known as the dihedral group of order 8, and is either denoted Dih 4, D 4 or D 8, depending on the convention. This was an example of a non-abelian group: the operation ∘ here is not commutative , which can be seen from the table; the table is not symmetrical about the main diagonal.
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 product of r ' s and f ' s. However, we have, for example, rfr = f −1, r 7 = r −1, etc., so such products are not unique in D 8. Each such product equivalence can be expressed ...
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 .
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.
A free group of rank k clearly has subgroups of every rank less than k. Less obviously, a (nonabelian!) free group of rank at least 2 has subgroups of all countable ranks. The commutator subgroup of a free group of rank k > 1 has infinite rank; for example for F(a,b), it is freely generated by the commutators [a m, b n] for non-zero m and n.
In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example (). They arise naturally in the study of arithmetic properties of quadratic forms and other classical topics in number theory .
A finitely generated group has a solvable word problem if and only if it can be embedded in every algebraically closed group. The proofs of these results are in general very complex. However, a sketch of the proof that a countable group C {\displaystyle C\ } can be embedded in an algebraically closed group follows.