Definition

A group is an ordered pair $(G, \cdot)$ consisting of a set $G$ and a binary operation $\cdot: G\times G\rightarrow G, (a,b)\mapsto a\cdot b$ such that the following are satisfied:

1. (Associativity) $(a\cdot b)\cdot c=a\cdot(b\cdot c)$ for all $a,b,c\in G$.
2. (Identity) There exists a unique $e\in G$ such that for every $x\in G$, $x\cdot e=e\cdot x=x$. The unique $e$ is called the identity of $G$.
3. (Inverse) For every $x\in G$, there exists some $y\in G$ such that $x\cdot y=y\cdot x=e$. Here $y$ is said to be the inverse of $x$, and is often denoted by $x^{-1}$.

By abuse of notation, we simply denote by $G$ for a group.