Normal Subgroups

As mentioned, when H < G, we have [G : H] as a set of disjoint left cosets. One question intrigues us : when does the set G : H form a group? Of course, first we need to define the operation. The most natural choice would then be (g1H) (g2H) ® (g1g2) H.

Does this definition make sense, i.e. if g1H = g1'H, g2H = g2'H, then (g1g2)H = (g1'g2')H? If it does, then the other axioms of identity, inverse, and associativity, follow naturally. The necessary condition for the definition to make sense:

(g1g2)-1 (g1'g2') = g2-1 (g1-1g1') g2 (g2-1g2') Î H Þ g2-1 (g1-1g1') g2 Î H.

Hence g-1hg Î H for all g Î G, h Î H.


We are thus inspired to give the following definition :

Definition
A subgroup H < G is said to be a normal subgroup if g-1hg Î H for all g Î G, h Î H.

The following theorem shows other equivalent definitions of normal subgroup.

Theorem 1

Let H < G be a subgroup. The following are equivalent :

  1. H is normal in G.
  2. For each gÎ G, gH = Hg.
  3. g-1Hg = H for all g Î G.
Here is the proof.


And now, the crux of the concept of normality : group quotient.

Theorem 2
If H < G is normal, then the set of left cosets G : H forms a group under the product : (g1H) (g2H) ® (g1g2)H.
This group is denoted by G / H.

The proof is provided here.

Also, we have the following result :

Theorem 3
The intersection of a collection of normal subgroups of G is also normal in G.

The proof is given here.


Normality is not transitive, i.e. it is in general not true that if H is normal in G, and K is normal in H, then K is normal in G. However, it is true that if K is normal in G, then K is normal in any subgroup of G which contains K. More generally, we have

Theorem 4
If N is normal in G, and H is a subgroup of G, then N Ç H is a normal subgroup of H.

Click here for the proof.