Please explain what's wrong with the proof that every group element is its own inverse.

Question

What is wrong with my proof here?

Proof:

Let $a, b$ be elements of a group and let $aa = b$.

Through manipulation, we see that $$a = ba^{-1}$$ $$b^{-1}a = a^{-1}$$ $$b^{-1}aa^{-1} = e$$ $$b^{-1} = e$$

We of course know that $bb^{-1} = e$ And from above, we see that $be = e$ or $b = e$. And since $aa = b$, we can see that $aa = e$ and that $a=a^{-1}$.

Answer 1

In going from

$$b^{-1}a = a^{-1}$$

to

$$b^{-1}aa^{-1} = e$$

we had already assumed the conclusion, because we implicitly used the "fact" that $(a^{-1})^2 = e$.

Answer 2

From $$ b^{-1}a=a^{-1}$$ multiplying by $a$ you get $$ ba^2=e, $$ not the line you got.