Normal subgroup

Definition

A subgroup H of a group G is termed normal if the following equivalent conditions are satisfied:

1. Given any $h\in H$ and $g\in G$, we have $ghg^{-1}\in H$
2. H occurs as the kernel of a homomorphism from G. In other words, there is a homomorphism $\phi :G\to K$ such that the inverse image of the identity element of K is H.
3. Every inner automorphism of G sends H to within itself
4. Every inner automorphism of G restricts to an automorphism of H

Some elementary examples and nonexamples

All subgroups in Abelian groups

In an Abelian group, every subgroup is normal. This is because if $G$ is an Abelian group, and $g,h\in G$, then $ghg^{-1}=h$.

More generally, any subgroup inside the center of a group is normal.

It is not, however, true that if every subgroup of a group is normal, then the group must be Abelian. A counterexample is the quaternion group.

All characteristic subgroups

A characteristic subgroup of a group is a subgroup which is invariant under all automorphisms of the whole group. Characteristic subgroups are normal, because normality requires invariance only under inner automorphisms, which are a particular kind of automorphism.

In particular, subgroups like the center, the commutator subgroup, the Frattini subgroup are examples of characteristic, and hence normal, subgroups.

A smallest non-example

The smallest example of a non-normal subgroup is a subgroup of order two in the symmetric group on three elements. Explicitly, we can take the cyclic subgroup of order two generated by the 2-cycle $(12)$ in the symmetric group of permutations on symbols $1,2,3$.

External links

• Normal subgroup on Mathworld
• Normal subgroup on Planetmath