Problem #WSP-000301

Problem

The set of symmetries of an object (e.g. a square) form an object called a group. We can formally define a group $G$ as follows.

A is a non-empty set $G$ with a binary operation $\ast$ satisfying the following axioms (you can think of them as rules). A binary operation takes two elements of $G$ and gives another element of $G$.

1. Closure: For all $g$ and $h$ in $G$, $g\ast h$ is also in $G$.

2. Identity: There is an element $e$ of $G$ such that $e\ast g=g=g\ast e$ for all $g$ in $G$.

3. Associativity: For all $g$, $h$ and $k$ in $G$, $(g\ast h)\ast k=g\ast (h\ast k)$.

4. Inverses: For every $g$ in $G$, there exists a $g^{-1}$ in $G$ such that $g\ast g^{-1}=e$.

Prove that the symmetries of the ‘reduce-reuse-recycle’ symbol form a group.