1.1 Review of Groups

Definition

Let $G$ be a set with a binary operation $G\times G\to G$. We say:

• $G$ is a semigroup if $a(bc)=(ab)c$;
• $G$ is a monoid if further there exists the two-sided identity element $e\in G$;
• $G$ is a group if further $\forall a\in G$, there exists $a^{-1}\in G$ such that $a{a}^{-1}=a^{-1}a=e$.