Group Homomorphism — Definition, Formula & Examples

A group homomorphism is a function from one group to another that respects the group operation — applying the operation before or after the function gives the same result.

Let $(G, \ast)$ and $(H, \cdot)$ be groups. A function $\phi: G \to H$ is a group homomorphism if for all $a, b \in G$ , $\phi(a \ast b) = \phi(a) \cdot \phi(b)$ .

Key Formula

\phi(a \ast b) = \phi(a) \cdot \phi(b) \quad \text{for all } a, b \in G

Where:

$\phi$ = The function from group G to group H
$G$ = The domain group with operation ∗
$H$ = The codomain group with operation ·
$a, b$ = Arbitrary elements of G

How It Works

To verify that a function is a group homomorphism, pick arbitrary elements

a, b

from the domain group, compute

\phi(a \ast b)

on one side and

\phi(a) \cdot \phi(b)

on the other, and check that they are equal. Two important consequences follow automatically: the homomorphism maps the identity of

G

to the identity of

H

, and it maps inverses to inverses, i.e.,

\phi(a^{-1}) = \phi(a)^{-1}

. You do not need to verify these separately — they are forced by the defining property.

Worked Example

Problem: Let G = (ℤ, +) and H = (ℤ, +). Show that ϕ(n) = 3n is a group homomorphism.

Set up the condition: We need to verify that ϕ(a + b) = ϕ(a) + ϕ(b) for all integers a and b.

\phi(a + b) = 3(a + b) = 3a + 3b

Compare both sides: Now compute ϕ(a) + ϕ(b) separately.

\phi(a) + \phi(b) = 3a + 3b

Conclude: Since 3(a + b) = 3a + 3b for all integers a and b, the homomorphism condition holds.

\phi(a + b) = \phi(a) + \phi(b) \; \checkmark

Answer: ϕ(n) = 3n is a group homomorphism from (ℤ, +) to (ℤ, +).

Why It Matters

Group homomorphisms are the central structure-preserving maps studied in abstract algebra. They underpin the First Isomorphism Theorem, which connects quotient groups to images of homomorphisms. In applications, they appear in coding theory, cryptography, and physics whenever symmetry groups need to be compared or simplified.

Common Mistakes

Mistake: Assuming a homomorphism must be bijective (one-to-one and onto).

Correction: A homomorphism only needs to preserve the operation. A bijective homomorphism has a special name — an isomorphism. Many useful homomorphisms are neither injective nor surjective.