Is $f(g,h) = g^2$ a binary operation on a group $G$?

Question

If $(G,\cdot)$ is a group, then $f : G \times G\to G$, defined by $f(g, h) = g^2$, is a binary operation on $G$. Is it true or false and why?

Answer 1

$f(a,f(b,c)) = f(a,b^2) = a^2$

$f(f(a,b),c) = f(a^2, c) = a^4$.

So probably not associative, depending on the $\cdot$ operation.