If $G$ is a group and $a,b\in G$ and $e$ neutral element why $ab = aeb$? The Problem I have if this would be true then one could also say $aaa^{-1}b=ab$.
Why do I know that there don't exist counterexamples for which the inequality holds?

$aaa^{-1}b = a(aa^{-1})b = aeb = (ae)b = ab$
So yes, you CAN say that, and you'd be one hundred percent correct when you do say that.

Why do I know that there don't exist counterexamples for which the inequality holds?

The same way you know that there aren't any counter examples to $k\times \frac 1k =1$ when $k \ne 0$.
BY DEFINITION $aa^{-1} = e$ and BY DEFINITION $aeb = (ae)b = ab$.  So that is a PROOF.
