Group

A Group is a set $G$ together with an operation $\ast$ on $G$ such that each of the following holds:

1. Closure: for all $\forall g,h\in G$, $g\ast h\in G$.
2. Existence of an identity element: $\exists e_G\in G$ such that for all $g\in G$, $g\ast e_G=e_G\ast g=g$.
3. Existence of inverse elements: For all $g\in G$ there exists an inverse element $g^{-1}\in G$ such that $g\ast g^{-1}=g^{-1}\ast g=e_G$.
4. Associativity: for all $g,h,k\in G$, $g\ast (h\ast k)=(g\ast h)\ast k$.
5. Rarely, if Abelian, Commutativity: for all $g,h\in G$, $g\ast h=h\ast g$.

Usually however, we don’t write the operation $\ast$, instead $g\ast h$ would be written as $gh$. By definition, $g^0:=e_G$ for all $g\in G$.

Given that a group $G$ follows these axioms for all $g\in G$, then that also gives the group six properties:

1. $G$ has only one identity.
2. $g$ has only one inverse.
3. $(g^{-1}{)}^{-1}=g$.
4. Let $h_1,h_2\in G$. If $g{h}_1=g{h}_2$ or $h_{1}g=h_{2}g$ then $h_1=h_2$.
5. $(h_1{h}_2{)}^{-1}=h_2^{-1}{h}_1^{-1}$.
6. $(g^n{)}^{-1}=g^{-n}$.