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Normal Subgroups, Quotients, and the Isomorphism Theorems

Quotient groups are how you forget what you do not care about. We revisit normality from the action viewpoint, then prove and apply the three isomorphism theorems — the lattice-level grammar that every later argument is written in.

Why normality is the right condition

A subgroup N of G is normal, written N ◁ G, when gNg⁻¹ = N for every g — equivalently, N is a union of conjugacy classes, equivalently the left and right cosets coincide. The point of normality is that it is exactly the condition under which the set of cosets G/N inherits a well-defined multiplication (aN)(bN) = (ab)N. Without normality this rule is ambiguous; with it, G/N becomes a genuine group, the quotient. Normal subgroups are precisely the kernels of homomorphisms out of G, which is why they matter at all. (This is the exact group-theoretic analogue of how an ideal lets you form a quotient ring.)

The three isomorphism theorems

These isomorphism theorems are the grammar of the subject. Internalize them so that you can quote them without thinking.

  1. First (the fundamental homomorphism theorem): if φ: G → H is a homomorphism, then G / ker φ ≅ im φ. Every quotient is an image and every image is a quotient — the two pictures of “collapsing” coincide.
  2. Second (the diamond): if H ≤ G and N ◁ G, then HN is a subgroup, H ∩ N ◁ H, and H / (H ∩ N) ≅ HN / N. It tells you how a subgroup sees a quotient.
  3. Third (quotient of a quotient): if N ≤ M are both normal in G, then (G/N) / (M/N) ≅ G/M. You may cancel N as if it were a common factor.
  4. Bonus (the correspondence theorem): subgroups of G/N correspond bijectively, and order-preservingly, to subgroups of G containing N, with normal matching normal. This is what lets you reason about the quotient by looking upstairs in G.
First isomorphism theorem in action: the sign homomorphism.

sgn : S_n -> {+1, -1},   sigma |-> +1 if even, -1 if odd.
This is a homomorphism onto the 2-element group.

  ker(sgn) = even permutations = A_n   (the alternating group).
  im(sgn)  = {+1, -1} = Z/2Z.

First isomorphism theorem:   S_n / A_n  ≅  Z/2Z.
Hence A_n is normal of index 2, and |A_n| = n!/2.

Another one-liner: the determinant det : GL(2,R) -> R*
is a homomorphism onto the nonzero reals.
  ker(det) = SL(2,R)   ⇒   GL(2,R) / SL(2,R) ≅ R*.
The first isomorphism theorem identifies A_n and SL as kernels in one line each.

The commutator subgroup and abelianization

Here is the cleanest application. The commutator subgroup G′ = [G, G] is generated by all commutators [a, b] = aba⁻¹b⁻¹. It is normal (even characteristic), and the quotient G/G′ is abelian. Better: G′ is the smallest normal subgroup with abelian quotient, so G/G′ — the abelianization — is the universal abelian image of G. Any homomorphism from G to an abelian group factors uniquely through G/G′. This is your first taste of a universal property, and it is exactly the object the solvable group machinery in guide 5 iterates.