Quotient group

If N is normal in G, the cosets of N form a group in their own right. Quotient groups are the main tool for breaking a group into smaller pieces.

Definition

Let N ⊴ G. The quotient group G/N is the set of left cosets {gN : g ∈ G} with operation

(gN)(hN) = (gh)N.

This is well-defined precisely because N is normal: if g₁N = g₂N and h₁N = h₂N, then g₁h₁N = g₂h₂N. The identity of G/N is N = eN, and (gN)⁻¹ = g⁻¹N.

Order. |G/N| = [G : N] = |G|/|N|.

Quotient map. π : G → G/N, g ↦ gN, is a surjective homomorphism with ker π = N.

Isomorphism theorems

For a homomorphism φ : G → H and subgroups M, N ≤ G:

First isomorphism theorem. G / ker φ ≅ im φ.
Second isomorphism theorem. If M ≤ G and N ⊴ G, then MN ≤ G, M ∩ N ⊴ M, and MN / N ≅ M / (M ∩ N).
Third isomorphism theorem. If N ⊴ M ⊴ G and N ⊴ G, then (G/N) / (M/N) ≅ G/M.
Correspondence theorem. Subgroups of G/N are in bijection with subgroups of G containing N, via H ↔ H/N. Normal subgroups correspond to normal subgroups.

The first is proved directly: g · ker φ ↦ φ(g) is a well-defined isomorphism G/ker φ → im φ.

Worked example: `S₄ / V₄ ≅ S₃`

Let V₄ = {e, (12)(34), (13)(24), (14)(23)} ⊴ S₄. Then |S₄/V₄| = 24/4 = 6.

Each coset corresponds to an unordered partition of {1,2,3,4} into two pairs. There are 3 such partitions — {12|34}, {13|24}, {14|23} — and S₄ permutes them faithfully. This gives a homomorphism S₄ → S₃ with kernel V₄, hence by the first isomorphism theorem S₄ / V₄ ≅ S₃. See S₄.

Examples

ℤ / nℤ ≅ Cₙ.
Sₙ / Aₙ ≅ C₂, via the sign homomorphism.
Dₙ / Cₙ ≅ C₂, the rotation/reflection dichotomy.
G / Z(G) cyclic ⟹ G abelian (classical exercise).
G / [G, G] is the abelianization — see commutator.

Computation

Polycyclic setting. A pc presentation of G/N is obtained by adding pc relations n = 1 for each generator of N, then reducing via collection. The result is again a pc group.
Permutation setting. If G acts on some set X with kernel N, the induced action gives G/N as a concrete permutation group on X.
Size of normal subgroup lattice. Normal subgroups correspond under the correspondence theorem to normal subgroups of quotients, so enumerating them reduces to iterated chief factor computations (see chief series).