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, and an inverse for every element — a subgroup is a smaller group sitting inside a larger one, and a homomorphism is a map that preserves the operation. If you remember one example, use the integers under addition: it carries the definition, a clean subgroup, and an easy homomorphism test.

When this framework applies

Group theory is the right lens whenever a problem has repeatable, reversible structure: symmetries of shapes, modular arithmetic, permutations, linear algebra, and parts of physics. The four axioms guarantee a system where you can combine elements, undo what you did, and trust that regrouping does not change the result — and that stability is precisely what makes groups appear in so many places. Think of each operation as a legal move: a group is a setting where moves combine, there is a do-nothing move, and every move can be reversed. You do not need advanced examples to benefit from this view. Even at a basic level, group theory helps you recognize when two problems that look unrelated on the surface actually share the same underlying structure, so a result proved once can be reused everywhere that structure appears.

The steps to check structure

1. Fix the operation. The same set behaves differently under different operations, so name it first.
2. Check the four group axioms:
   1. Closure: if $a,b \in G$ then $ab \in G$.
   2. Associativity: $(ab)c = a(bc)$ for all $a,b,c$.
   3. Identity: some $e \in G$ with $ea = ae = a$.
   4. Inverses: each $a$ has $a^{-1} \in G$ with $aa^{-1} = a^{-1}a = e$. If even one fails, it is not a group.
3. Test a subgroup carefully. A subset is a subgroup only if it uses the same operation and still satisfies all the requirements inside the larger group.
4. Check preservation for a map. For a homomorphism $f$, confirm $f(ab) = f(a)f(b)$ for all elements.

Full example: integers, even integers, and parity

The group $(\mathbb{Z}, +)$. Take $\mathbb{Z} = \{\dots,-2,-1,0,1,2,\dots\}$ under addition. Closure: a sum of integers is an integer. Associativity: $(a+b)+c = a+(b+c)$. Identity: $0$, since $a+0 = a$. Inverses: $-a$, since $a+(-a) = 0$. So $(\mathbb{Z}, +)$ is a group.

A subgroup. The even integers $2\mathbb{Z} = \{\dots,-4,-2,0,2,4,\dots\}$ form a subgroup: adding two evens gives an even, $0$ is even (identity present), and the negative of an even is even. A smaller closed world where the same operation still works.

A homomorphism. 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 $a,b \in G$, and when the operation is addition this reads $f(a+b) = f(a) + f(b)$. The point is that combining first and then mapping must agree with mapping first and then combining. Define $f : \mathbb{Z} \to \mathbb{Z}_2$ by $f(n) = n \bmod 2$, where $\mathbb{Z}_2 = \{0,1\}$ with addition mod $2$; this simply records whether an integer is even or odd. To test it, compare the parity of a sum, $f(a+b)$, against the sum of the parities, $f(a)+f(b)$. For $a=3$, $b=5$: $f(3) = 1$, $f(5) = 1$, $f(3+5) = f(8) = 0$, and $f(3)+f(5) = 1+1 = 0 \pmod 2$. The two routes agree for all integers, so $f(a+b) = f(a)+f(b)$ — the map preserves the operation.

Where you get stuck, and how to verify

• Forgetting the operation is part of the data. "The integers form a group" is incomplete: true under addition, false under multiplication (most integers lack a multiplicative inverse in $\mathbb{Z}$).
• Assuming every subset is a subgroup. It must keep the identity, stay closed, and contain inverses. The positive integers are not a subgroup of $(\mathbb{Z}, +)$ — no $0$, no additive inverses.
• Treating a homomorphism as any function. Its whole job is preserving the operation; if $f(ab) = f(a)f(b)$ fails, it is not one.
• Mixing notation across groups. Addition in one, multiplication in another, composition in a third — use the correct operation on each side.

Practice it yourself

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

holds modulo $3$. For one more step, ask the same questions about rotations of an equilateral triangle — often where group theory starts to feel like a tool rather than a definition, because there the elements are concrete actions you can picture, yet they still obey the very same four axioms as the integers under addition.