Generating set of a group

In abstract algebra, a generating set of a group G is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses.

Related Topics:
Abstract algebra - Group - Subset

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More generally, if S is a subset of a group G, then , the subgroup generated by S, is the smallest subgroup of G containing every element 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.

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

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When there is only a single element x in S, is usually written as . In this case, 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 it has order |G|, or that equals the entire group G.

Related Topics:
Cyclic group - Order

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If S is finite, then a group G =  is called finitely generated. The structure of finitely generated abelian groups in particular is easily described. Many theorems that are true for finitely generated groups fail for groups in general.

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Every finite group is finitely generated since  = G. The integers under addition are an example of an infinite group which is finitely generated by both and , but the group of rationals under addition cannot be finitely generated. No uncountable group can be finitely generated.

Related Topics:
Integer - Rationals - Uncountable

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Different subsets of the same group can be generating subsets; for example, if p and q are integers with gcd(p, q) = 1, then also generates the group of integers under addition.

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The most general group generated by a set S is the group freely generated by S. Every group generated by S is isomorphic to a factor group of this group, a feature which is utilized in the expression of a group's presentation.

Related Topics:
'''freely generated''' - Isomorphic - Factor group - Presentation

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An interesting companion topic is that of non-generators. An element x of the group G is a non-generator if every set S containing x that generates G, still generates G when x is removed from S. In the integers with addition, the only non-generator is 0. The set of all non-generators forms a subgroup of G, the Frattini subgroup.

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