Normal Subgroups — Mathematics Knowledge System

Overview

A normal subgroup is a subgroup N of a group G that is invariant under conjugation by every element of G: gNg⁻¹ = N for all g ∈ G. Normal subgroups are exactly the subgroups whose left and right cosets coincide, which is precisely the condition needed to make the set of cosets G/N into a group under a well-defined operation. They were introduced by Évariste Galois in his study of when polynomial equations are solvable by radicals, and every kernel of a group homomorphism is a normal subgroup, and conversely.

Intuition

A subgroup is normal when it looks the same 'from every viewpoint' in G — relabeling the group by conjugating with any element g doesn't move N at all, it just permutes N's own elements among themselves. This is exactly the property needed so that multiplying cosets, (aN)(bN) = (ab)N, gives a consistent answer regardless of which representatives a and b you pick; without normality, different representatives could give different coset products, so the quotient construction would break.

Formal Definition

Definition

Let G be a group and N ≤ G a subgroup. N is normal in G, written N ⊴ G, if any of the following equivalent conditions hold:

gNg^{-1} = N \quad \text{for all } g \in G

Conjugation invariance

gN = Ng \quad \text{for all } g \in G

Left cosets equal right cosets

gng^{-1} \in N \quad \text{for all } g \in G,\ n \in N

Elementwise conjugation stays in N

N = \ker(\varphi) \text{ for some homomorphism } \varphi: G \to H

Normal subgroups are exactly kernels

Worked Examples

Let G be abelian and H ≤ G. For any g ∈ G and h ∈ H, conjugation gives:
$ghg^{-1} = hgg^{-1} = h$
Since gh = hg in an abelian group, conjugation fixes every element of H exactly.
$gHg^{-1} = H$

Answer: In an abelian group every subgroup is normal, since conjugation is always trivial (ghg⁻¹ = h).

Practice Problems

Difficulty 4/10

Explain why the trivial subgroup {e} and the whole group G are always normal in G.

Difficulty 5/10

Prove that any subgroup H of G with index [G:H] = 2 is normal.

Difficulty 5/10

In D₄ (dihedral group of order 8, symmetries of a square), the rotation subgroup R = {e, r, r², r³} has index 2. Is R normal in D₄?

Quiz

Summary

N ⊴ G means gNg⁻¹ = N for all g ∈ G, equivalently left cosets equal right cosets (gN = Ng).
Every subgroup of an abelian group is normal; any subgroup of index 2 is automatically normal.
Normal subgroups are exactly the kernels of group homomorphisms, and are precisely what's needed to build the quotient group G/N.

References

WebsiteWikipedia — Normal subgroup