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normal subgroup (Definition)

A subgroup $ H$ of a group $ G$ is normal if $ aH = Ha$ for all $ a \in G$. Equivalently, $ H \subset G$ is normal if and only if $ aHa^{-1} = H$ for all $ a \in G$, i.e., if and only if each conjugacy class of $ G$ is either entirely inside $ H$ or entirely outside $ H$.

The notation $ H \trianglelefteq G$ or $ H \triangleleft G$ is often used to denote that $ H$ is a normal subgroup of $ G$.

The kernel $ \ker(f)$ of any group homomorphism $ f: G \longrightarrow G'$ is a normal subgroup of $ G$. More surprisingly, the converse is also true: any normal subgroup $ H \subset G$ is the kernel of some homomorphism (one of these being the projection map $ \rho: G \longrightarrow G/H$, where $ G/H$ is the quotient group).



"normal subgroup" is owned by djao.
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See Also: quotient group, normalizer

Other names:  normal
Also defines:  normality

Attachments:
normality of subgroups is not transitive (Example) by yark
a subgroup of index 2 is normal (Theorem) by alozano
one-sided normality of subsemigroup (Definition) by lars_h
the kernel of a group homomorphism is a normal subgroup (Theorem) by alozano
normality of subgroups of prime index (Theorem) by azdbacks4234
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Cross-references: quotient group, projection map, converse, group homomorphism, kernel, conjugacy class, group, subgroup
There are 131 references to this entry.

This is version 7 of normal subgroup, born on 2002-01-05, modified 2007-07-04.
Object id is 1305, canonical name is NormalSubgroup.
Accessed 18604 times total.

Classification:
AMS MSC: 20A05 (Group theory and generalizations :: Foundations :: Axiomatics and elementary properties)
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