$$\begin{align*} & {\color{blue}{(S \;, \;\#)}}\;\; \text{ : is a Abelian Group if it satisfies all the following properties } \\ \\ &\begin{cases} & 1.\text{ Closure} \\ & 2.\text{ Associative} \\ & 3.\text{ Identity element exits} \\ & 4.\text{ Inverse for all elements} \in S \\ & 5.\text{ Commutative} \\ \end{cases} \\ \\ \hline \\ &\text{ A } {\color{red}{\bf \text{group}}} \text{ satisfies properties } \;\; {\color{blue}{1,2,3,4}} \\ &\text{ When a group } \;\; {\color{blue}{(S \;, \;\#)}}\;\; \text{satisfies commutative property} \\ &\Rightarrow {\color{blue}{(S \;, \;\#)}}\;\; \text{is } {\color{red}{\bf \text{Abelian}}} \text{ also.} \\ \\ \hline \\ \end{align*}$$

$\begin{align*} &\Rightarrow \text{LHS} = \left ( a \; \# \; b \right )^2 \\ &\Rightarrow \text{LHS} = \left ( a \; \# \; b \right ) \# \left ( a \; \# \; b \right ) \\ &\Rightarrow \text{LHS} = a \; \# \; \left (\bf b \; \# \; a \right ) \; \# \; b \\\\ \hline \\ &\Rightarrow \text{RHS} = a^2 \; \# \; b^2 \\ &\Rightarrow \text{RHS} = \left ( a \; \# \; a \right ) \# \left ( b \; \# \; b \right ) \\ &\Rightarrow \text{RHS} = a \; \# \; \left (\bf a \; \# \; b \right ) \; \# \; b \\\\ \hline \\ &\text{Given LHS = RHS} \\ &\Rightarrow \left ( b \; \# \; a \right ) = \left ( a \; \# \; b \right ) \\ &\Rightarrow \text{Given group follows commutative property }\\ &\Rightarrow \text{Given group is} {\color{red}{\bf \text{Abelian}}} & \\ \end{align*}$