Group Theory — Groups, Subgroups & Homomorphisms

Group theory explains when a set and an operation fit together in a stable way. A group has four ingredients: closure, associativity, an identity element, and an inverse for every element. A subgroup is a smaller group inside a larger one, and a homomorphism is a map that preserves the operation.

If you only remember one example, use the integers under addition. It shows the group definition, gives a clean subgroup, and makes the homomorphism idea easy to test.

Group definition: the four axioms

A group is a set $G$ together with an operation, often written multiplicatively as $ab$, such that four conditions hold:

1. Closure: if $a,b \in G$, then $ab \in G$.
2. Associativity: $(ab)c = a(bc)$ for all $a,b,c \in G$.
3. Identity: there is an element $e \in G$ such that $ea = ae = a$ for every $a \in G$.
4. Inverses: for each $a \in G$, there is an element $a^{-1} \in G$ such that $aa^{-1} = a^{-1}a = e$.

That is the full definition. If even one condition fails, the set with that operation is not a group.

Why the definition matters

The definition gives you a system where you can combine elements, undo what you did, and trust that regrouping does not change the result. That is why groups show up in symmetry, modular arithmetic, permutations, and matrix algebra.

If you think of an operation as a legal move, then a group is a system where legal moves can be combined, there is a do-nothing move, and every move can be reversed.

Example: why $(\mathbb{Z}, +)$ is a group

Take the set of all integers $\mathbb{Z} = \{\dots,-2,-1,0,1,2,\dots\}$ with the operation of addition.

This is a group:

• Closure holds because the sum of two integers is still an integer.
• Associativity holds because $(a+b)+c = a+(b+c)$ for integers.
• The identity element is $0$ because $a+0 = 0+a = a$.
• The inverse of $a$ is $-a$ because $a + (-a) = 0$.

So $(\mathbb{Z}, +)$ is a group.

This example is the right starting point because it also makes subgroups and homomorphisms concrete.

Subgroup definition with the even integers

A subgroup is a subset of a group that is itself a group under the same operation.

Inside $(\mathbb{Z}, +)$, consider the even integers:

$$2\mathbb{Z} = \{\dots,-4,-2,0,2,4,\dots\}$$

This is a subgroup of $\mathbb{Z}$ under addition because:

• adding two even integers gives another even integer
• $0$ is even, so the identity is still there
• the negative of an even integer is still even

So $2\mathbb{Z}$ is not just a subset. It keeps the same algebraic rules as the larger group.

That is the main idea of a subgroup: it is a smaller closed world where the same operation still works.

Homomorphism definition: preserving the operation

A homomorphism is a function between groups that preserves the operation.

If $f : G \to H$ is a homomorphism, then

$$f(ab) = f(a)f(b)$$

for all $a,b \in G$.

The exact symbols depend on the groups involved. If the operation is addition, the same condition is often written as

$$f(a+b) = f(a) + f(b)$$

The point is the same: combine first and then map, or map first and then combine. A homomorphism makes those two routes agree.

Worked example: parity map from $\mathbb{Z}$ to $\mathbb{Z}_2$

Define $f : \mathbb{Z} \to \mathbb{Z}_2$ by

$$f(n) = \text{the remainder of } n \text{ mod } 2$$

Here $\mathbb{Z}_2 = \{0,1\}$ with addition modulo $2$.

This function records whether an integer is even or odd. To check that it is a homomorphism, compare both sides:

$$f(a+b)$$

is the parity of the sum, while

$$f(a) + f(b)$$

adds the two parities mod $2$.

Those match for all integers $a$ and $b$, so

$$f(a+b) = f(a) + f(b)$$

in $\mathbb{Z}_2$.

For example, if $a=3$ and $b=5$, then

$$f(3) = 1,\qquad f(5) = 1,\qquad f(3+5) = f(8) = 0$$

and

$$f(3)+f(5) = 1+1 = 0 \pmod{2}$$

So the map preserves the group operation.

Common mistakes in group theory

Forgetting that the operation is part of the data

Saying "the integers form a group" is incomplete unless the operation is clear. The integers form a group under addition, but not under multiplication, because most integers do not have multiplicative inverses inside $\mathbb{Z}$.

Assuming every subset is a subgroup

A subset has to keep the identity, stay closed under the operation, and contain inverses. For example, the positive integers are not a subgroup of $(\mathbb{Z}, +)$ because they do not contain $0$ and they do not contain additive inverses.

Treating homomorphisms like arbitrary functions

A homomorphism is not just any map between sets. Its whole job is to preserve the operation. If that condition fails, it is not a group homomorphism.

Mixing notation across different groups

In one group the operation may be addition, in another multiplication, and in another composition. The homomorphism rule must use the correct operation on each side.

Where group theory is used

Group theory is used whenever a problem has repeatable reversible structure. Common examples include symmetries of shapes, modular arithmetic, permutations, linear algebra, and parts of physics.

You do not need advanced examples to benefit from it. Even at a basic level, group theory helps you recognize when different problems share the same underlying structure.

Try a similar problem

Start with $(\mathbb{Z}, +)$. Check whether the multiples of $3$ form a subgroup. Then test the map $g(n) = n \bmod 3$ from $\mathbb{Z}$ to $\mathbb{Z}_3$ and verify whether

$$g(a+b) = g(a) + g(b)$$

holds modulo $3$.

If you want one more step, try the same questions with rotations of an equilateral triangle. That is often where group theory starts to feel like a tool instead of a definition.