generating set of a group
Let be a group.
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 .
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|
|Date of creation||2013-03-22 15:37:14|
|Last modified on||2013-03-22 15:37:14|
|Last modified by||yark (2760)|
|Defines||subgroup generated by|
|Defines||closed under inverses|
|Defines||group generated by|