Group Homomorphisms and Isomorphisms: Concepts and Examples

A group homomorphism is a map φ from a group G to a group H satisfying φ(ab) = φ(a)φ(b) for all a, b in G: multiply-then-map gives the same result as map-then-multiply. This single equation is the whole idea — a homomorphism carries the multiplication structure of one group into another.

Homomorphisms are how we compare groups. An isomorphism — a bijective homomorphism — means two groups are structurally identical even when their elements look different; for instance, ℤ and the even integers are isomorphic via n ↦ 2n. Without homomorphisms, group theory would be isolated examples; with them, it becomes a connected landscape we can classify.

Key Concepts

• Homomorphism: φ: G → H with φ(ab) = φ(a)φ(b).
• Kernel: ker φ = {a ∈ G : φ(a) = e_H}; always a normal subgroup of G, measuring how much φ collapses.
• Image: im φ = {φ(a) : a ∈ G}, a subgroup of H.
• Isomorphism: bijective homomorphism (G ≅ H). Automorphism: isomorphism G → G; endomorphism: homomorphism G → G.
• Useful facts: φ(e_G) = e_H; φ(a^{-1}) = φ(a)^{-1}; φ(a^n) = φ(a)^n; φ is injective iff ker φ = {e_G}; the order of φ(a) divides the order of a.

Formulas

• Defining identity: φ(ab) = φ(a)φ(b) for all a, b ∈ G. (Abstract groups; no physical units.)
• Identities and inverses: φ(e_G) = e_H; φ(a^{-1}) = φ(a)^{-1}; φ(a^n) = φ(a)^n.
• Order dividing: if a has finite order m, the order of φ(a) divides m — it can be strictly smaller.

Worked Example

Consider φ: ℤ → ℤ/6ℤ, φ(n) = n mod 6. It is a homomorphism: φ(m + n) = (m + n) mod 6 = φ(m) + φ(n). Numerically: φ(14) = 2 (14 = 6·2 + 2); φ(−7) = 5 (−7 = 6·(−2) + 5); φ(6) = 0. The kernel is 6ℤ, the multiples of 6; the image is all of ℤ/6ℤ. φ is not injective (φ(6) = φ(0) = 0) but is surjective — every remainder class is hit.

Common Misconceptions

• "A homomorphism must be bijective." No — the constant map to the identity is a valid homomorphism that is neither injective nor surjective.
• "The kernel is always just {e}." Only for injective maps; the kernel measures exactly how far φ is from being injective.
• "Any map between groups is a homomorphism." The equation must hold: n ↦ n + 1 on ℤ fails since φ(2 + 3) = 6 ≠ 7 = φ(2) + φ(3).

Real-World Applications

• Cryptography: the discrete logarithm problem (maps x ↦ g^x) underlies Diffie–Hellman key exchange.
• Physics and chemistry: symmetry groups are compared via homomorphisms; group representations are homomorphisms into matrix groups.
• Error detection: parity checks are homomorphisms (ℤ/2ℤ)^n → ℤ/2ℤ.
• Number theory: reduction mod n, the homomorphism ℤ → ℤ/nℤ, opens up modular arithmetic.

A homomorphism preserves the group operation, its kernel measures the collapsing, and a bijective homomorphism — an isomorphism — certifies two groups are the same structure in different clothing.