How to prove that it is a group? Let $G$ be a set with a binary operation *, associating to each pair of elements $x$ and $y$ of $G$ a third element $x*y$ of $G$. Suppose that the following properties are satisfied:


*

*$(x*y)*z = x*(y*z)$ for all elements $x$, $y$, and $z$ of $G$ (the Associative Law);

*there exists an element $e$ of $G$ such that $e*x = x$ for all elements $x$ of $G$;

*for each element $x$ of $G$ there exists an element $x'$ of $G$ satisfying $x'*x = e$.


How to prove
i. $x*e=x$ and
ii. $x*x'=e$,
so that $G$ is a group?

$x'(xe)=(x'x)e=ee=e=x'x$, $x''x'xe=x''x'x$, $xe=x$ takes care of i. 
