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 ifevery element of can be expressed asa 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
inversesif 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 elementis called a cyclic group.A group that has a generating set with only finitely many elementsis 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 isthe minimum cardinality of a generating set of .(This is sometimes just called the rank of , but this cancause 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 |