generating set of a group
Let G be a group.
A subset X⊆G is said to generate G
(or to be a generating set of G)
if no proper subgroup
of G contains X.
A subset X⊆G generates G if and only if
every element of G can be expressed as
a product of elements of X and inverses
of elements of X
(taking the empty product to be the identity element
).
A subset X⊆G is said to be closed under
inverses
if x-1∈X whenever x∈X;
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 group.
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 subgroup 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 | Generator |
Defines | generate |
Defines | generates |
Defines | generated by |
Defines | subgroup generated by |
Defines | generating rank |
Defines | closed under inverses |
Defines | group generated by |