normal subgroup

A subgroupMathworldPlanetmathPlanetmath H of a group G is normal if aH=Ha for all aG. Equivalently, HG is normal if and only if aHa-1=H for all aG, i.e., if and only if each conjugacy classMathworldPlanetmathPlanetmath of G is either entirely inside H or entirely outside H.

The notation HG or HG is often used to denote that H is a normal subgroupMathworldPlanetmath of G.

The kernel ker(f) of any group homomorphismMathworldPlanetmath f:GG is a normal subgroup of G. More surprisingly, the converseMathworldPlanetmath is also true: any normal subgroup HG is the kernel of some homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (one of these being the projection map ρ:GG/H, where G/H is the quotient groupMathworldPlanetmath).

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 NormalizerMathworldPlanetmath
Defines normality