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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, 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 .
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.
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 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 ...
Group extension is usually described as a hard problem; it is termed the extension problem. To consider some examples, if G = K × H {\displaystyle G=K\times H} , then G {\displaystyle G} is an extension of both H {\displaystyle H} and K {\displaystyle K} .
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study, in universal algebra one takes the class of groups as an object of study.
A more sophisticated definition of an algebraic group over a field is that it is that of a group scheme over (group schemes can more generally be defined over commutative rings). Yet another definition of the concept is to say that an algebraic group over k {\displaystyle k} is a group object in the category of algebraic varieties over k ...
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