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单词 GeneratingSetOfAGroup
释义

generating set of a group


Let G be a group.

A subset XG is said to generate G(or to be a generating setPlanetmathPlanetmath of G)if no proper subgroupMathworldPlanetmath of G contains X.

A subset XG generates G if and only ifevery element of G can be expressed asa productMathworldPlanetmathPlanetmathPlanetmath of elements of X and inversesMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of elements of X(taking the empty product to be the identity elementMathworldPlanetmath).A subset XG is said to be closed underPlanetmathPlanetmath inversesif x-1X whenever xX;if a generating set X of G is closed under inverses,then every element of G is a product of elements of X.

A group that has a generating set with only one elementis called a cyclic groupMathworldPlanetmath.A group that has a generating set with only finitely many elementsis called a finitely generated group.

If X is an arbitrary subset of G,then the subgroupMathworldPlanetmathPlanetmath of G generated by X, denoted by X,is the smallest subgroup of G that contains X.

The generating rank of G isthe minimum cardinality of a generating set of G.(This is sometimes just called the rank of G, but this cancause confusion with other meanings of the term rank.)If G is uncountable, then its generating rank is simply |G|.

Titlegenerating set of a group
Canonical nameGeneratingSetOfAGroup
Date of creation2013-03-22 15:37:14
Last modified on2013-03-22 15:37:14
Owneryark (2760)
Last modified byyark (2760)
Numerical id7
Authoryark (2760)
Entry typeDefinition
Classificationmsc 20A05
Classificationmsc 20F05
Synonymgenerating set
Related topicPresentationgroup
Related topicGeneratorPlanetmathPlanetmathPlanetmathPlanetmath
Definesgenerate
Definesgenerates
Definesgenerated by
Definessubgroup generated by
Definesgenerating rank
Definesclosed under inverses
Definesgroup generated by
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更新时间:2025/5/4 16:01:43