Groups and Their Axioms

A group is a set $G$ with a single binary operation $*$ satisfying four axioms:

1. Closure: $a * b \in G$ for all $a, b \in G$.
2. Associativity: $(a * b) * c = a * (b * c)$.
3. Identity: there is $e \in G$ with $e * a = a * e = a$ for all $a$.
4. Inverses: each $a \in G$ has some $a^{-1} \in G$ with $a * a^{-1} = a^{-1} * a = e$.

Commutativity ($a*b = b*a$) is not required. A group that happens to commute is called abelian.

Worked example. $(\mathbb{Z}, +)$ is a group: closed, associative, identity $0$, and the inverse of $n$ is $-n$. But $(\mathbb{Z}, \times)$ is not a group — most integers (like $2$) have no multiplicative inverse inside $\mathbb{Z}$. A rich family of finite groups is $\mathbb{Z}_n$, the integers $\{0,1,\dots,n-1\}$ under addition mod $n$.