In good book called: "An Introduction: Theory of Spinors" defines the group as follows:
A non-empty set $G$ of elements $a$, $b$, $c$,..., such as numbers, mappings, transformations, is called a group if the following axioms are satisfied:
1.There exists an operation in the set $G$ which associates to each two elements $a$ and $b$ of $G$ a third element $c$ of $G$. This operation is called multiplication, and the element $c$ is called the product of $a$ and $b$, denoted by $c=ab$;
2.The multiplication is associative, namely, If $a$, $b$ and $c$ are elements of $G$, then $(ab)c=a(bc)$;
3.The set $G$ contains a right identity, namely, there exists an element $e$ such that $ae=a$ for each element $a$ of $G$; and
4.For each element $a$ of $G$ there exists a right inverse element, denoted by $a^{-1}$, such that $aa^{-1}=e$.
The book put exercise to show that the right identity is also left identity.
My problem with this is that the proof I know requires having left inverse but the definition above defines right inverse and doesn't prove left inverse yet, see the following proof for showing the right identity is left identity:
Let $a$ be any element in $G$ and write $x=ea$. Then $a^{-1}x=a^{-1}(ea)=(a^{-1}e)a=a^{-1}a=e$. Then solving for $x$ we obtain $x=ae=a$.
In the above proof we used left inverse while it is not given yet.
