To prove set is a group Given a non empty set  together with associative binary operation $*$ on $G$ such that 
$a*x=b$ and $y*a=b$ have solutions in $G$ for all $a,b$ in $G$ 
To prove it is a group 
Hints to get started 
Thanks

The following equations have solutions for every $a$ and $b$ :
$$\begin{align} ax&=b\tag{R_1(a,b)}\\
 xa&=b \tag{R_2(a,b)} \end{align}$$
Given an arbitrary element $a$, let $e$ be a solution of $ax=a$ (solution of $R_1(a,a)$). Then for any $b$ there exists $y$ such that $ya=b$ (use $R_2(a,b)$), hence $be=yae=ya=b$. So $e$ is a right unit. Given an arbitrary element $t$ the equation $tx=e$ (use $R_1(t,e)$) has always solution, so $G$ has a right unit and every element has a right inverse. Then it is a group.
