Homomorphism and Isomorphism of Groups

Definition

Let $(G, *)$ and $(H, \circ)$ be groups. A function $f: G \to H$ is called a group homomorphism if for all $a, b \in G$,

$$f(a * b) = f(a) \circ f(b).$$

This means the group operation is preserved under the function.

A group homomorphism $f: G \to H$ is called an isomorphism if it is both:

Injective

• (one-to-one), and

Surjective

• (onto).

If such a function exists, then $G$ and $H$ are called isomorphic groups, written as

$$G \cong H.$$

This means the two groups have the same algebraic structure.

Main Content

1. Group Homomorphism

• A group homomorphism preserves the group operation, so the image of a product is the product of the images.
• It also preserves important group properties such as identity and inverses:
• $f(e_G) = e_H$, where $e_G$ and $e_H$ are identity elements.
• $f(a^{-1}) = [f(a)]^{-1}$ for every $a \in G$.

A homomorphism may compress a group into a smaller group, combine several elements into the same image, or map a group into a substructure of another group. It does not have to be one-to-one or onto.

Example 1:
Define $f: (\mathbb{Z}, +) \to (\mathbb{Z}_n, +)$ by

$$f(x) = x \bmod n.$$

Then for any integers $a, b$,

$$f(a+b) = (a+b) \bmod n = (a \bmod n) + (b \bmod n) = f(a) + f(b).$$

So $f$ is a homomorphism.

Example 2:
Let $f: (\mathbb{R}, +) \to (\mathbb{R}^{+}, \times)$ be defined by

$$f(x) = e^x.$$

Then

$$f(x+y) = e^{x+y} = e^x e^y = f(x)f(y),$$

so this is also a homomorphism.

2. Kernel and Image of a Homomorphism

• The kernel of a homomorphism $f: G \to H$ is the set $$\ker(f) = \{g \in G : f(g) = e_H\}.$$ It measures which elements collapse to the identity in $H$.

• The image of $f$ is $$\operatorname{Im}(f) = \{f(g) : g \in G\}.$$ It is the set of all outputs of the map and forms a subgroup of $H$.

The kernel is a very important concept because it tells us how much information is lost under the homomorphism. If the kernel contains only the identity element, then the homomorphism is injective.

A fundamental result is:

$$f \text{ is injective } \iff \ker(f) = \{e_G\}.$$

Example:
For the map $f: \mathbb{Z} \to \mathbb{Z}_n$, $f(x)=x \bmod n$,

$$\ker(f) = n\mathbb{Z} = \{\ldots, -2n, -n, 0, n, 2n, \ldots\}.$$

This means all multiples of $n$ map to the identity element in $\mathbb{Z}_n$.

3. Group Isomorphism

• An isomorphism is a homomorphism that preserves structure perfectly and has a reverse map that is also a homomorphism.
• If $f: G \to H$ is an isomorphism, then every group-theoretic property of $G$ is shared by $H$, such as:
• size of the group,
• commutativity,
• order of elements,
• subgroup structure patterns.

An isomorphism tells us that two groups are essentially the same, even if the elements are written differently.

Example 1:
The groups $(\mathbb{Z}_4, +)$ and $(\{1, i, -1, -i\}, \times)$ are isomorphic.
A possible isomorphism is:

$$0 \mapsto 1,\quad 1 \mapsto i,\quad 2 \mapsto -1,\quad 3 \mapsto -i.$$

This preserves the operation because addition mod 4 corresponds exactly to multiplication of fourth roots of unity.

Example 2:
The group $(\mathbb{R}, +)$ is isomorphic to $(\mathbb{R}^{+}, \times)$ via $f(x)=e^x$.
This shows that addition of real numbers and multiplication of positive real numbers have the same group structure under this mapping.

ASCII Diagram for How a Homomorphism Preserves Structure

G:   a ---- b ---- a*b
      |      |       |
      f      f       f
      v      v       v
H:  f(a) -- f(b) -- f(a)*f(b)

This shows that applying the function after combining elements gives the same result as combining their images.

Working / Process

1. Check the mapping rule

• Identify the function $f: G \to H$.
• Verify whether it preserves the group operation: $$f(a*b) = f(a)\circ f(b)$$ for all elements $a, b \in G$.

2. Find the identity and inverse behavior

• Confirm that the identity of the first group maps to the identity of the second group.
• Check whether inverses are preserved: $$f(a^{-1}) = [f(a)]^{-1}.$$

3. Determine whether it is an isomorphism

• Check injectivity using the kernel: $$\ker(f)=\{e_G\}$$ if and only if the map is one-to-one.

• Check surjectivity by verifying whether every element of $H$ is hit by the map.

• If both conditions hold, the homomorphism is an isomorphism.

Advantages / Applications

• Homomorphisms help simplify complicated groups by mapping them into more manageable ones while preserving structure.
• Isomorphisms allow mathematicians to classify groups by identifying when two seemingly different groups are actually the same in structure.
• These concepts are widely used in many areas of mathematics and science, including:
• symmetry analysis in geometry,
• cryptography,
• number theory,
• representation theory,
• solving equations using modular arithmetic.

Summary

• Group homomorphisms preserve the group operation.
• Isomorphisms are homomorphisms that are both one-to-one and onto.
• Kernel and image are central tools for understanding homomorphisms.
• Important terms to remember: homomorphism, isomorphism, kernel, image, injective, surjective, identity element, inverse, isomorphic groups.