normal subgroup (original) (raw)
A subgroup
H of a group G is normal if aH=Ha for all a∈G. Equivalently, H⊂G is normal if and only if aHa-1=H for all a∈G, i.e., if and only if each conjugacy class
of G is either entirely inside H or entirely outside H.
The notation H⊴G or H◁G is often used to denote that H is a normal subgroup of G.
The kernel ker(f) of any group homomorphism f:G⟶G′ is a normal subgroup of G. More surprisingly, the converse
is also true: any normal subgroup H⊂G is the kernel of some homomorphism
(one of these being the projection map ρ:G⟶G/H, where G/H 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 |