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A generator, in category theory, is an object that can be used to distinguish morphisms; In topology, a collection of sets that generate the topology is called a subbase; Generating set of a topological algebra: S is a generating set of a topological algebra A if the smallest closed subalgebra of A containing S is A
WolframAlpha gathers data from academic and commercial websites such as the CIA's The World Factbook, the United States Geological Survey, a Cornell University Library publication called All About Birds, Chambers Biographical Dictionary, Dow Jones, the Catalogue of Life, [3] CrunchBase, [6] Best Buy, [7] and the FAA to answer queries. [8]
By definition, the group is cyclic if and only if it has a generator g (a generating set {g} of size one), that is, the powers ,,, …, give all possible residues modulo n coprime to n (the first () powers , …, give each exactly once).
The 5th roots of unity in the complex plane form a group under multiplication. Each non-identity element generates the group. In abstract algebra, a generating set of a group is a subset of the group set such that every element of the group can be expressed as a combination (under the group operation) of finitely many elements of the subset and their inverses.
In abstract algebra, a cyclic group or monogenous group is a group, denoted C n (also frequently n or Z n, not to be confused with the commutative ring of p-adic numbers), that is generated by a single element. [1]
There is an obvious semigroup homomorphism j : S → G(S) that sends each element of S to the corresponding generator. This has a universal property for morphisms from S to a group: [ 12 ] given any group H and any semigroup homomorphism k : S → H , there exists a unique group homomorphism f : G → H with k = fj .