A group is a pair $(G,\cdot)$ where $G$ is a set and $\cdot$ is a binary operation on elements of $G$ such that: 1. Closure: $$g,h\in G\implies g\cdot h\in G$$ 2. Associativity: $$(g\cdot h)\cdot k=g\cdot(h\cdot k)$$ 3. Existence of Identity: $$\exists1:1\cdot g=g$$ 4. Existence of Inverse: $\forall g\in G,\exists g^{-1}\in G$ such that $$g^{-1}\cdot g=1$$