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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.
Plus teacher and student package: Group Theory This package brings together all the articles on group theory from Plus, the online mathematics magazine produced by the Millennium Mathematics Project at the University of Cambridge, exploring applications and recent breakthroughs, and giving explicit definitions and examples of groups.
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 ...
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 ...
In mathematics, a structure on a set (or on some sets) refers to providing it (or them) with certain additional features (e.g. an operation, relation, metric, or topology). Τhe additional features are attached or related to the set (or to the sets), so as to provide it (or them) with some additional meaning or significance.
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 .
In mathematics, the category Grp (or Gp [1]) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category . The study of this category is known as group theory .
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.