First Isomorphism Theorem: Explained with Examples

The First Isomorphism Theorem is the single most useful theorem in group theory: for any homomorphism φ: G → H, G / ker φ ≅ im φ. Every homomorphism is secretly an isomorphism once you divide out the redundancy (the kernel) and restrict to the output (the image).

This turns every homomorphism into a classification tool. Instead of studying φ directly, you study the clean isomorphism between a quotient group and a subgroup — the engine behind the classification of cyclic groups and the construction of modular arithmetic.

Key Concepts

• Kernel as normal subgroup: ker φ is always normal in G, so G / ker φ always exists.
• Cosets: the elements of G / ker φ are cosets a·ker φ; a·ker φ = b·ker φ exactly when φ(a) = φ(b).
• Canonical map: ψ(a·ker φ) = φ(a) gives the isomorphism G / ker φ → im φ; it is well defined because φ agrees on each coset, injective because ψ(a·ker φ) = e forces a ∈ ker φ, and onto im φ by construction.

Formulas

• The theorem: G / ker φ ≅ im φ via a·ker φ ↦ φ(a). (Abstract groups; no physical units.)
• Order corollary: |im φ| = |G| / |ker φ| for finite groups.
• Special cases: ker φ = {e} gives G ≅ im φ; im φ = H gives G / ker φ ≅ H.
• Limit: the theorem describes only the image; it says nothing about elements of H outside im φ.

Worked Example

Take φ: ℤ → ℤ, φ(n) = 2n. It is a homomorphism: φ(m + n) = 2(m + n) = φ(m) + φ(n). The kernel is {n : 2n = 0} = {0}; the image is the even integers 2ℤ. The theorem gives ℤ / {0} ≅ 2ℤ, i.e. ℤ ≅ 2ℤ — the integers are isomorphic to the even integers via n ↦ 2n. Numeric check: φ(5) = 10, φ(−3) = −6, and distinct inputs give distinct outputs because the kernel is trivial. A "twice as sparse" set can be structurally identical to the whole.

Common Misconceptions

• "G ≅ H for every homomorphism φ: G → H." False — the theorem gives G / ker φ ≅ im φ. You must divide by the kernel.
• "The quotient G / ker φ always exists." It exists because kernels are always normal; quotient by an arbitrary subgroup is not a group.
• "a·ker φ ↦ φ(a) might depend on the representative." It does not: if b = a·k with k ∈ ker φ, then φ(b) = φ(a)φ(k) = φ(a). This well-definedness is the heart of the proof.

Real-World Applications

• Cyclic groups: every infinite cyclic group is ≅ ℤ; every cyclic group of order n is ≅ ℤ/nℤ.
• Linear algebra: rank–nullity is the vector-space cousin: dim(im T) = dim V − dim(ker T).
• Modular arithmetic: building ℤ/nℤ from ℤ via n ↦ n mod n is the prototype of cryptography's arithmetic.
• Ring theory: the same theorem holds for rings (R / ker φ ≅ im φ).

The First Isomorphism Theorem says every homomorphism factors into a collapse (quotient by the kernel) followed by a perfect identification (isomorphism onto the image) — a universal template for understanding structure-preserving maps.