Normal subgroup

A normal subgroup is a subgroup stable under conjugation. Normality is exactly the condition that lets you form a quotient group and exactly the condition satisfied by kernels of homomorphisms.

Definition

A subgroup N ≤ G is normal, written N ⊴ G, if

g N g⁻¹ ⊆ N   for all g ∈ G.

The inclusion is automatically equality, since applying the condition to g⁻¹ gives the reverse.

Equivalent characterizations

For a subgroup N ≤ G the following are equivalent:

1. g N g⁻¹ = N for every g ∈ G.
2. gN = Ng for every g (left cosets coincide with right cosets).
3. N is a union of conjugacy classes of G.
4. N = ker φ for some homomorphism φ out of G.
5. The set of left cosets G/N carries a well-defined group operation (see quotient).

Characterization 4 is the most useful: normal subgroups are precisely the kernels.

Examples

• Trivial. {e} and G are normal in G.
• Every subgroup of an abelian group is normal.
• Index-2 subgroups. If [G : H] = 2 then H ⊴ G, since both cosets (H and G ∖ H) are forced. Hence Aₙ ⊴ Sₙ and the rotation subgroup Cₙ ⊴ Dₙ.
• Center. Z(G) ⊴ G, and more generally every subgroup of Z(G) is normal.
• Commutator subgroup. [G, G] ⊴ G, and any subgroup containing [G, G] is normal.
• Normal subgroups of S₄: {e}, V₄ = {e, (12)(34), (13)(24), (14)(23)}, A₄, S₄.

Non-examples

• {e, (12)} ≤ S₃ is not normal: (13)(12)(13) = (23) ∉ {e, (12)}.

Properties

• An intersection of normal subgroups is normal.
• A normal subgroup times any subgroup is a subgroup: if N ⊴ G and M ≤ G, then NM = {nm : n ∈ N, m ∈ M} is a subgroup; if both are normal, so is NM.
• Normality is not transitive: K ⊴ H and H ⊴ G need not give K ⊴ G. For example ⟨(12)(34)⟩ ⊴ V₄ ⊴ S₄, yet ⟨(12)(34)⟩ is not normal in S₄.
• The normal subgroups of a group form a modular sub-lattice of its subgroup lattice.
• A quotient G/N is not fixed by the isomorphism type of N: in G = C₄ × C₂ the normal subgroups ⟨(2,0)⟩ and ⟨(0,1)⟩ are both ≅ C₂, yet the quotients are C₂ × C₂ and C₄.
• A quotient G/N need not embed in G: Q₈ / Z(Q₈) ≅ V₄, but every order-4 subgroup of Q₈ is cyclic.
• Schur–Zassenhaus: a normal Hall subgroup N ⊴ G always has a complement, so G = N ⋊ K is a semidirect product with |K| = [G : N], and the complements are all conjugate.
• A normal Hall subgroup is unique and characteristic: it is the only subgroup of its order.
• In the subgroup lattice, the Frattini subgroup is the meet of all maximal subgroups, and the socle is the join of all minimal normal subgroups.

See also

• Characteristic subgroup — the strengthening where stability is required under every automorphism, not just inner ones.
• Conjugacy class — normality = stability under conjugation.
• Quotient group — the construction unlocked by normality.
• Homomorphism — kernels are normal.
• Center, commutator subgroup — canonical normal subgroups.
• Simple group — groups with no non-trivial normal subgroups.