Symmetric Group on 3 Letters/Group Presentation

Group Presentation of Symmetric Group on $3$ Letters
The group presentation of the symmetric group on $3$ letters is given by:
 * $S_3 := \gen {a, b: a^3 = b^2 = \paren {a b}^2 = e}$

Hence:
 * $\begin{array}{c|cccccc}

& e    & a     & a^2   & b     & a b   & a^2 b \\ \hline e    & e     & a     & a^2   & b     & a b   & a^2 b \\ a    & a     & a^2   & e     & a b   & a^2 b & b     \\ a^2  & a^2   & e     & a     & a^2 b & b     & a b   \\ b    & b     & a^2 b & a b   & e     & a^2   & a     \\ a b  & a b   & b     & a^2 b & a     & e     &  a^2  \\ a^2 b & a^2 b & a b  & b     & a^2   & a     & e     \\ \end{array}$

Proof
Let $G = \gen {a, b: a^3 = b^2 = \paren {a b}^2 = e}$.

It is to be demonstrated that $S_3$ is isomorphic to $G$.

Consider the Cayley table for $S_3$:

We demonstrate that $S_3$ is isomorphic to $G$ as follows:

Let $\phi: G \to S_3$ be a mapping that sends:
 * $\phi: a \mapsto p$
 * $\phi: b \mapsto r$

We have:
 * $p^3 = e$
 * $r^2 = e$
 * $\paren {p r}^2 = s^2 = e$

demonstrating that $S_3$ has the same group presentation as $G$.

Hence the result.