The homomorphism $\psi$ takes $G$ to $G$, where $G$ is a group, and let $b=|G|$ (i.e., the number of elements in $G$). Let $a$ be the number of elements $e$ in $G$ such that $\psi(e)=e^2$. Find when $\frac{a}{b}=\frac{3}{100}$? I also was interested in the largest possible value of $\frac{a}{b}$ besides $1$?
I tried simulating this on my own for very small groups but couldn't get anywhere. The second question is out of curiosity, I'm not sure it is possible to find out.