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Topic 3 of 8

Homomorphisms & Isomorphisms

A homomorphism is a map between groups that respects the group operation. Homomorphisms reveal how groups are related to each other — and the first isomorphism theorem gives a precise sense in which every homomorphism is secretly a quotient map. Isomorphisms tell you when two groups are structurally identical, even if they look different.

Key Concepts

Homomorphism

A map

\phi: G \to H

satisfying

\phi(ab) = \phi(a)\phi(b)

for all

a,b \in G

. Automatically sends identity to identity and inverses to inverses.

Kernel

\ker(\phi) = \{g \in G : \phi(g) = e_H\}

. Always a normal subgroup of

G

. A homomorphism is injective iff

\ker(\phi) = \{e\}

Image

\text{im}(\phi) = \{\phi(g) : g \in G\}

. Always a subgroup of

H

Isomorphism

A bijective homomorphism.

G \cong H

means

G

and

H

are structurally identical — same multiplication table, same element orders.

Automorphism

An isomorphism

\phi: G \to G

from a group to itself. The set

\text{Aut}(G)

of all automorphisms forms a group.

|\text{Aut}(\mathbb{Z}_n)| = \varphi(n)

(Euler's totient).

Key Equations

First Isomorphism Theorem

G / \ker(\phi) \;\cong\; \mathrm{im}(\phi)

Every homomorphism factors through the quotient by its kernel.

Order Relation

|G| = |\ker(\phi)| \cdot |\mathrm{im}(\phi)|

For finite groups, the kernel and image sizes multiply to

|G|

Homomorphisms

\mathbb{Z}_m \to \mathbb{Z}_n

|\mathrm{Hom}(\mathbb{Z}_m, \mathbb{Z}_n)| = \gcd(m, n)

The number of distinct homomorphisms between cyclic groups.

Worked Example

Kernel of a Cyclic Homomorphism

Problem

Let $\phi: \mathbb{Z}_{12} \to \mathbb{Z}_3$ be defined by $\phi(n) = n \bmod 3$ . Find $|\ker(\phi)|$ and verify the first isomorphism theorem.

Solution

$\ker(\phi) = \{n \in \mathbb{Z}_{12} : n \equiv 0 \pmod 3\} = \{0, 3, 6, 9\}$ .

$|\ker(\phi)| = 4$ . The image is $\text{im}(\phi) = \mathbb{Z}_3$ , so $|\text{im}(\phi)| = 3$ .

Check: $|\ker|\times|\text{im}| = 4\times3 = 12 = |\mathbb{Z}_{12}|$ . ✓

First isomorphism theorem: $\mathbb{Z}_{12}/\ker(\phi) \cong \mathbb{Z}_3$ , i.e. $\mathbb{Z}_{12}/\{0,3,6,9\} \cong \mathbb{Z}_3$ . ✓

Answer

|\ker(\phi)| = 4

Practice

Exercises

6 problems

1 of 6

For $\phi: \mathbb{Z}_{18} \to \mathbb{Z}_6$ defined by $\phi(n) = n \bmod 6$ , what is $|\ker(\phi)|$ ?

Your answer

2 of 6

The number of homomorphisms $\mathbb{Z}_m \to \mathbb{Z}_n$ equals $\gcd(m,n)$ . How many homomorphisms are there from $\mathbb{Z}_8$ to $\mathbb{Z}_{12}$ ?

Your answer

3 of 6

$|\mathrm{Aut}(\mathbb{Z}_n)| = \varphi(n)$ (Euler's totient). Compute $|\mathrm{Aut}(\mathbb{Z}_{10})|$ . Recall $\varphi(10) = \varphi(2)\varphi(5) = 1\times4$ .

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4 of 6

If $\phi: G \to H$ is an isomorphism and $|G| = 120$ , what is $|H|$ ?

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5 of 6

For $\phi: \mathbb{Z}_{20} \to \mathbb{Z}_5$ defined by $\phi(n) = n \bmod 5$ , find $|\ker(\phi)|$ .

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6 of 6

Since 7 is prime, $\varphi(7) = 7 - 1 = 6$ . What is $|\mathrm{Aut}(\mathbb{Z}_7)|$ ?

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Key Takeaways

A homomorphism $\phi: G \to H$ respects the group operation: $\phi(ab)=\phi(a)\phi(b)$ . The kernel is always a normal subgroup.
First isomorphism theorem: $G/\ker(\phi) \cong \text{im}(\phi)$ . Every homomorphism is (up to isomorphism) a quotient map.
Two groups are isomorphic ( $G\cong H$ ) iff there is a bijective homomorphism between them — they are structurally the same group.
$|\text{Hom}(\mathbb{Z}_m, \mathbb{Z}_n)| = \gcd(m,n)$ and $|\text{Aut}(\mathbb{Z}_n)| = \varphi(n)$ are fundamental counting results.