normal closure
Let S be a subset of a group G.
The normal closure of S in G is the intersection of all normal subgroups
of G that contain S, that is
⋂S⊆N⊴GN. |
The normal closure of S is the smallest normal subgroup of G that contains S, and so is also called the normal subgroup generated by S.
It is not difficult to show that the normal closure of S is the subgroup generated by all the conjugates of elements of S.
The normal closure of S in G is variously denoted by ⟨SG⟩ or ⟨S⟩G or SG.
If H is a subgroup of G, and H is of finite index in its normal closure, then H 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 |