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 if every element of G can be expressed as a 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 inverses if 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 element is called a cyclic groupMathworldPlanetmath. A group that has a generating set with only finitely many elements is 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 is the minimum cardinality of a generating set of G. (This is sometimes just called the rank of G, but this can cause confusion with other meanings of the term rank.) If G is uncountable, then its generating rank is simply |G|.

Title generating set of a group
Canonical name GeneratingSetOfAGroup
Date of creation 2013-03-22 15:37:14
Last modified on 2013-03-22 15:37:14
Owner yark (2760)
Last modified by yark (2760)
Numerical id 7
Author yark (2760)
Entry type Definition
Classification msc 20A05
Classification msc 20F05
Synonym generating set
Related topic Presentationgroup
Related topic GeneratorPlanetmathPlanetmathPlanetmathPlanetmath
Defines generate
Defines generates
Defines generated by
Defines subgroup generated by
Defines generating rank
Defines closed under inverses
Defines group generated by