normal closure
Let be a subset of a group . The normal closure of in is the intersection of all normal subgroups of that contain , that is
The normal closure of is the smallest normal subgroup of that contains , and so is also called the normal subgroup generated by .
It is not difficult to show that the normal closure of is the subgroup generated by all the conjugates of elements of .
The normal closure of in is variously denoted by or or .
If is a subgroup of , and is of finite index in its normal closure, then is said to be nearly normal.
Title | normal closure |
Canonical name | NormalClosure1 |
Date of creation | 2013-03-22 14:41:50 |
Last modified on | 2013-03-22 14:41:50 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 9 |
Author | yark (2760) |
Entry type | Definition |
Classification | msc 20A05 |
Synonym | normal subgroup generated by |
Synonym | conjugate closure |
Synonym | smallest normal subgroup containing |
Related topic | Normalizer |
Related topic | CoreOfASubgroup |
Defines | nearly normal |