generating set of a group
Let be a group.
A subset is said to generate
(or to be a generating set of )
if no proper subgroup
of contains .
A subset generates if and only if
every element of can be expressed as
a product of elements of and inverses
of elements of
(taking the empty product to be the identity element
).
A subset is said to be closed under
inverses
if whenever ;
if a generating set of is closed under inverses,
then every element of is a product of elements of .
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 is an arbitrary subset of ,
then the subgroup of generated by , denoted by ,
is the smallest subgroup of that contains .
The generating rank of is the minimum cardinality of a generating set of . (This is sometimes just called the rank of , but this can cause confusion with other meanings of the term rank.) If is uncountable, then its generating rank is simply .
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 |