What is a Group?

A group is one of the simplest structures in abstract algebra, and one of the most useful. It asks a minimal question: what is the essential shape of “doing something and undoing it”?

The definition

A group is a set $G$ together with a binary operation $\cdot$ (read: “combined with”) that satisfies four axioms:

1. Closure. For any $a, b \in G$, the result $a \cdot b$ is also in $G$. The operation stays inside the set.

2. Associativity. For any $a, b, c \in G$: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$. Grouping does not matter.

3. Identity. There exists an element $e \in G$ such that $e \cdot a = a \cdot e = a$ for all $a \in G$. There is a “do-nothing” element.

4. Inverses. For every $a \in G$, there exists an element $a^{-1} \in G$ such that $a \cdot a^{-1} = a^{-1} \cdot a = e$. Every action can be undone.

That is the entire definition. Four axioms. The surprising fact is how much follows from them.

Building intuition

Before the symbols settle, consider a concrete example: the integers under addition $(\mathbb{Z}, +)$.

• Closure: adding two integers gives an integer. ✓
• Associativity: $(1 + 2) + 3 = 1 + (2 + 3) = 6$. ✓
• Identity: $0 + n = n + 0 = n$ for all $n$. The identity is $0$. ✓
• Inverses: for any integer $n$, the inverse is $-n$, since $n + (-n) = 0$. ✓

So $(\mathbb{Z}, +)$ is a group. This is not surprising. But consider what does not work: the natural numbers $\mathbb{N} = \{0, 1, 2, \ldots\}$ under addition fail the inverse axiom. There is no natural number $m$ such that $3 + m = 0$.

A finite example: $\mathbb{Z}/4\mathbb{Z}$

The integers modulo 4, written $\mathbb{Z}/4\mathbb{Z}$ or $\mathbb{Z}_4$, are the set $\{0, 1, 2, 3\}$ with addition modulo 4. The Cayley table:

Every row and column is a permutation of $\{0,1,2,3\}$ — a hallmark of groups.

Worked example: verifying a group

Claim: The set $\{1, -1\}$ under multiplication forms a group.

• Closure: $1 \cdot 1 = 1$, $1 \cdot (-1) = -1$, $(-1)(-1) = 1$. All results are in $\{1, -1\}$. ✓
• Associativity: multiplication of real numbers is associative. ✓
• Identity: $1 \cdot a = a$ for all $a$. The identity is $1$. ✓
• Inverses: $1^{-1} = 1$ and $(-1)^{-1} = -1$. Each element is its own inverse. ✓

This is a group of order 2 — the smallest non-trivial group. It describes the symmetry of an object with exactly one non-trivial symmetry (a reflection with no rotation).

What groups are not

Some near-misses worth knowing:

• Monoid: satisfies closure, associativity, identity — but not necessarily inverses. Example: $(\mathbb{N}, +)$.
• Semigroup: only closure and associativity. Example: positive integers under addition.
• Abelian group: a group where additionally $a \cdot b = b \cdot a$ for all $a, b$. The integers under addition are abelian; the symmetries of a cube are not.