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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. If G = ⟨S⟩, then we say that S generates G, and the elements in S are called generators or group generators. If S is the empty set, then ⟨S⟩ is the trivial group {e}, since we consider the empty product to be the identity.

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rdf:type	programming language
rdfs:label	Generating set of a group (en)
rdfs:comment	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. If G = ⟨S⟩, then we say that S generates G, and the elements in S are called generators or group generators. If S is the empty set, then ⟨S⟩ is the trivial group {e}, since we consider the empty product to be the identity. (en)
sameAs	Generating set of a group
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Subject	Group theory
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gold:hypernym	Subset
prov:wasDerivedFrom	http://en.wikipedia.org/wiki/Generating_set_of_a_group?oldid=1061703562&ns=0
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has abstract	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 other words, if S is a subset of a group G, then ⟨S⟩, the subgroup generated by S, is the smallest subgroup of G containing every element of S, which is equal to the intersection over all subgroups containing the elements of S; equivalently, ⟨S⟩ is the subgroup of all elements of G that can be expressed as the finite product of elements in S and their inverses. (Note that inverses are only needed if the group is infinite; in a finite group, the inverse of an element can be expressed as a power of that element.) If G = ⟨S⟩, then we say that S generates G, and the elements in S are called generators or group generators. If S is the empty set, then ⟨S⟩ is the trivial group {e}, since we consider the empty product to be the identity. When there is only a single element x in S, ⟨S⟩ is usually written as ⟨x⟩. In this case, ⟨x⟩ is the cyclic subgroup of the powers of x, a cyclic group, and we say this group is generated by x. Equivalent to saying an element x generates a group is saying that ⟨x⟩ equals the entire group G. For finite groups, it is also equivalent to saying that x has order \|G\|. If G is a topological group then a subset S of G is called a set of topological generators if ⟨S⟩ is dense in G, i.e. the closure of ⟨S⟩ is the whole group G. (en)
foaf:isPrimaryTopicOf	http://en.wikipedia.org/wiki/Generating_set_of_a_group
is Wikipage redirect of	Generator (group) Generator (groups) Generator (monoid) Generator (semigroup) Group generated Group generator Group generators Cyclic subgroup
is Link from a Wikipage to another Wikipage of	Fox derivative Grigorchuk group Group (mathematics) Local rigidity Locally finite group Magnetic space group Normal closure (group theory) Normal form for free groups and free product of groups Analysis of flows Babai's problem Baby-step giant-step Banach–Tarski paradox Basis Baumslag–Solitar group Bianchi classification Birch and Swinnerton-Dyer conjecture Blancmange curve Monothetic group Monster group Monstrous moonshine Bracket Bracket (mathematics) Braid group Capped grope Cayley graph

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