$\begin{array}{lllc} \mathrlap{\rule[-2ex]{45em}{0.1ex}} & \text{Non-homomorphism} & \text{Cases which 'work'} & \phi(\text{identity})=\text{identity}? \\ \\ 1. & \adjust \begin{aligned}\phi\colon(\C^*,\times) & \to(\C,+)\\z & \mapsto z+\Re(z)\end{aligned} & \phi(z_1z_2)=\phi(z_1)+\phi(z_2)$ if $z_1=z_2=2 & No\\ \\ 2. & \adjust \begin{aligned}\phi\colon(\C^*,\times) & \to(\C^*,\times)\\z & \mapsto z\Re(z)\end{aligned} & \phi(z_1z_2)=\phi(z_1)\phi(z_2)$ if $z_1=z_2=1 & Yes\\ \\ 3. & \adjust \begin{aligned}\phi\colon(\C^*,\times) & \to(\C^*,\times)\\z & \mapsto e^{iz}\end{aligned} & \phi(z_1z_2)=\phi(z_1)\phi(z_2)$ if $z_1=z_2=2 & No\\ \\ 4. & \adjust \begin{aligned}\phi\colon(\R,+) & \to(\R,+)\\x & \mapsto \sin(x)\end{aligned} & \phi(x_1+x_2)=\phi(x_1)+\phi(x_2)$ if $x_2=n\pi & Yes\\ \\ 5. & \adjust \begin{aligned}\phi\colon(\R,+) & \to(\R,+)\\x & \mapsto 2^x\end{aligned} & \phi(x_1+x_2)=\phi(x_1)+\phi(x_2)$ if $x_1=x_2=1 & No\\ \\ 6. & \adjust \begin{aligned}\phi\colon(\R^*,\times) & \to(\R^*,\times)\\x & \mapsto 2^x\end{aligned} & \phi(x_1x_2)=\phi(x_1)\phi(x_2)$ if $x_1=x_2=2 & No\\ \\ 7. & \adjust \begin{aligned}\phi\colon(\R^*,\times) & \to(\R,+)\\x & \mapsto x\end{aligned} & \phi(xy)=\phi(x)+\phi(y)$ if $y=\frac{x}{x-1}\text{ eg }x=y=2 & No\\ \\ 8. & G\text{ a group of matrices}& \phi(AB)=\phi(A)\phi(B)$ if $A, B\text{ commute} & Yes\\ &\text{ under multiplication}\\&\begin{aligned}\phi\colon G & \to G\\A & \mapsto A^2\end{aligned} \\ \\ 9. & G\text{ a group of matrices}& \phi(A+B)=\phi(A)+\phi(B)$ if $A=B=0 \text{ or }& Yes\\ &\text{ under addition} & B=-A\text{ has an odd number of rows/columns} \\ &\begin{aligned}\phi\colon G & \to (\R,+)\\A & \mapsto \det(A)\end{aligned} \\ \\ 10. & \adjust \begin{aligned}\phi\colon S(\triangle) & \to S(\triangle)\\ & g\mapsto g^{-1}\end{aligned} & \phi(g_1\circ g_2)=\phi(g_1)\circ\phi(g_2)$ if $g_1,g_2\text{ commute} & Yes \end{array}$