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Non-homomorphismCases which ’work’ϕ(identity)=identity?1.ϕ ⁣:(C∗,×)→(C,+)z↦z+Re(z)ϕ(z1z2)=ϕ(z1)+ϕ(z2) if z1=z2=2No2.ϕ ⁣:(C∗,×)→(C∗,×)z↦zRe(z)ϕ(z1z2)=ϕ(z1)ϕ(z2) if z1=z2=1Yes3.ϕ ⁣:(C∗,×)→(C∗,×)z↦eizϕ(z1z2)=ϕ(z1)ϕ(z2) if z1=z2=2No4.ϕ ⁣:(R,+)→(R,+)x↦sin⁡(x)ϕ(x1+x2)=ϕ(x1)+ϕ(x2) if x2=nπYes5.ϕ ⁣:(R,+)→(R,+)x↦2xϕ(x1+x2)=ϕ(x1)+ϕ(x2) if x1=x2=1No6.ϕ ⁣:(R∗,×)→(R∗,×)x↦2xϕ(x1x2)=ϕ(x1)ϕ(x2) if x1=x2=2No7.ϕ ⁣:(R∗,×)→(R,+)x↦xϕ(xy)=ϕ(x)+ϕ(y) if y=xx−1 eg x=y=2No8.G a group of matricesϕ(AB)=ϕ(A)ϕ(B) if A,B commuteYes under multiplicationϕ ⁣:G→GA↦A29.G a group of matricesϕ(A+B)=ϕ(A)+ϕ(B) if A=B=0 or Yes under additionB=−A has an odd number of rows/columnsϕ ⁣:G→(R,+)A↦det⁡(A)10.ϕ ⁣:S(△)→S(△)g↦g−1ϕ(g1∘g2)=ϕ(g1)∘ϕ(g2) if g1,g2 commuteYes\begin{array}{lllc} \mathrlap{\rule[-2ex]{45em}{0.1ex}} & \text{Non-homomorphism} & \text{Cases which 'work'} & \phi(\text{identity})=\text{identity}? \\ \\ 1. & \begin{aligned}\phi\colon(\mathbb{C}^*,\times) & \to(\mathbb{C},+)\\z & \mapsto z+ \text{Re}(z)\end{aligned} & \phi(z_1z_2)=\phi(z_1)+\phi(z_2) \text{ if }z_1=z_2=2 & \text{No}\\ \\ 2. & \begin{aligned}\phi\colon(\mathbb{C}^*,\times) & \to(\mathbb{C}^*,\times)\\z & \mapsto z \text{Re}(z)\end{aligned} & \phi(z_1z_2)=\phi(z_1)\phi(z_2) \text{ if }z_1=z_2=1 & \text{Yes}\\ \\ 3. & \begin{aligned}\phi\colon(\mathbb{C}^*,\times) & \to(\mathbb{C}^*,\times)\\z & \mapsto e^{iz}\end{aligned} & \phi(z_1z_2)=\phi(z_1)\phi(z_2) \text{ if }z_1=z_2=2 & \text{No}\\ \\ 4. & \begin{aligned}\phi\colon(\mathbb{R},+) & \to(\mathbb{R},+)\\x & \mapsto \sin(x)\end{aligned} & \phi(x_1+x_2)=\phi(x_1)+\phi(x_2) \text{ if }x_2=n\pi & \text{Yes}\\ \\ 5. & \begin{aligned}\phi\colon(\mathbb{R},+) & \to(\mathbb{R},+)\\x & \mapsto 2^x\end{aligned} & \phi(x_1+x_2)=\phi(x_1)+\phi(x_2) \text{ if }x_1=x_2=1 & \text{No}\\ \\ 6. & \begin{aligned}\phi\colon(\mathbb{R}^*,\times) & \to(\mathbb{R}^*,\times)\\x & \mapsto 2^x\end{aligned} & \phi(x_1x_2)=\phi(x_1)\phi(x_2) \text{ if }x_1=x_2=2 & \text{No}\\ \\ 7. & \begin{aligned}\phi\colon(\mathbb{R}^*,\times) & \to(\mathbb{R},+)\\x & \mapsto x\end{aligned} & \phi(xy)=\phi(x)+\phi(y) \text{ if }y=\frac{x}{x-1}\text{ eg }x=y=2 & \text{No}\\ \\ 8. & G\text{ a group of matrices}& \phi(AB)=\phi(A)\phi(B) \text{ if }A, B\text{ commute} & \text{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) \text{ if }A=B=0 \text{ or }& \text{Yes}\\ &\text{ under addition} & B=-A\text{ has an odd number of rows/columns} \\ &\begin{aligned}\phi\colon G & \to (\mathbb{R},+)\\A & \mapsto \det(A)\end{aligned} \\ \\ 10. & \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) \text{ if }g_1,g_2\text{ commute} & \text{Yes} \end{array}