Example: Examples of Properties of Group Homomorphisms

Verifying the property (1) $f(e_G)=e_H$:
- The neutral element of $G=(\mathbb Z,+)$ is $e_G=0$ and the same neutral element has $H=(\{3n\mid n\in\mathbb Z\}, +)$, i.e. $e_H=0$. Thus $$f(0)=3\cdot 0=0.$$
Verifying the property (2) $f(x^{-1})=f(x)^{-1}$ for all $x\in G$:
- Let $x\in(\mathbb Z,+)$. Then $x^{-1}=-x.$
- Then $f(-x)=3\cdot(-x)=-3x$ which is the inverse element of $3x\in (\{3n\mid n\in\mathbb Z\},+).$

Verifying the property (1) $f(e_G)=e_H$:
- The neutral element of $G=(\mathbb R,+)$ is $e_G=0$ and the neutral element of $H=(\mathbb R^{ +}, \cdot)$ is $e_H=1$. We verify by exponential function zero $$\exp(0)=1.$$
Verifying the property (2) $f(x^{-1})=f(x)^{-1}$ for all $x\in G$:
- Let $x\in(\mathbb R,+)$. Then $x^{-1}=-x.$ We verify by the reciprocity law for the exponential function. $$\exp(-x)=\frac{1}{\exp(x)},$$ which is the inverse element of $\exp(x)\in (\mathbb R^{ +},\cdot).$

Verifying the property (1) $f(e_G)=e_H$:
- The neutral element of $G=(\mathbb R,+)$ is $e_G=0$ and the neutral element of $H=(\operatorname{GL}(2,\mathbb R),\cdot)$ is $$e_H=\pmatrix{1&0\\0&1}.$$ We verify for the in the example defined group homomorphism $$\rho(0)=\pmatrix{\cos(0)&-\sin(0)\\\sin(0)&\cos(0)}=\pmatrix{1&0\\0&1}.$$
Verifying the property (2) $f(x^{ -1})=f(x)^{ -1}$ for all $x\in G$:
- Let $x\in(\mathbb R,+)$. Then $x^{ -1}=-x.$ We verify by the eveness of cosine and the oddness of sine. $$\rho(-x)=\pmatrix{\cos( -x)&-\sin( -x)\\\sin( -x)&\cos( -x)}=\pmatrix{\cos(-x)&\sin(x)\\-\sin(x)&\cos(x)},$$ which is the inverse element of $H.$ This can be verified by the Pythagorean identity $$\pmatrix{\cos(x)&-\sin(x)\\\sin(x)&\cos(x)}\cdot \pmatrix{\cos(x)&\sin(x)\\-\sin(x)&\cos(x)}=\pmatrix{1&0\\0&1}.$$

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