Let $*$ be the operation $a*b = ab^2$ over the integers. Is this a group?

Question

I'm trying to prove associativity and I think this operation fails.

This is what I have: $$\begin{align} (a*b)*c &= a*(b*c)\\ (ab^2)*c &= a*(bc^2)\\ (ab^2)(c^2) &= (a)(bc^2)^2\\ ab^2c^2 &\neq ab^2c^4 \end{align}$$

Answer 1

Yes, you're correct. Well done!

Let $a=b=1, c=2$. That would complete things.

It would be a better proof, though, in my opinion, if you put sentences or at least punctuation marks between the equations. (This is just better form.)