normal subgroup
A subgroup of a group is normal if for all . Equivalently, is normal if and only if for all , i.e., if and only if each conjugacy class of is either entirely inside or entirely outside .
The notation or is often used to denote that is a normal subgroup of .
The kernel of any group homomorphism is a normal subgroup of . More surprisingly, the converse is also true: any normal subgroup is the kernel of some homomorphism (one of these being the projection map , where is the quotient group).
Title | normal subgroup |
---|---|
Canonical name | NormalSubgroup |
Date of creation | 2013-03-22 12:08:07 |
Last modified on | 2013-03-22 12:08:07 |
Owner | djao (24) |
Last modified by | djao (24) |
Numerical id | 11 |
Author | djao (24) |
Entry type | Definition |
Classification | msc 20A05 |
Synonym | normal |
Related topic | QuotientGroup |
Related topic | Normalizer |
Defines | normality |