Proving uniqueness (basics of group theory)

Question

If $(G,*)$ is a group, prove that the identity and the inverse elements are unique.

What I did for the first one is:

Suppose $\exists e,g\in G$ such that $\forall a\in G a*e=e*a=a$ and also that $\forall a\in G a*g=g*a=a$, then $g=g*e=e$, hence $g=e$. The first equallity happens because $e$ is an identity, and the second happens because $g$ is an identity.

Is this right? I checked some notes of another class and their proof is much more longer and complicated (although I understand every step), what am I doing wrong?

For the second one, can I use that if $a$ is the inverse of $b$, then $b$ is the inverse of $a$? I feel that I shouldn't, but I'm not sure why.

Answer 1

Your proof about the uniqueness of the identity element is fine.

For uniqueness of inverses, it's not a bad idea to say: let $b$ and $c$ both be inverses for $a$. Then $ab = e$ and $ac = e$, so $ab = ac$. Now multiply by $b$ on the left on each side.

Answer 2

For the first let $e'$ is also an identity element then $$e=e*e'=e'$$

and for the second if $a'$ and $a''$ are inverse of $a$ then $$a'aa''=(a'a)a''=a''=a'(aa'')=a'$$