Example: Examples of Kernels and Images of Group Homomorphisms

Kernel $\operatorname{ker}(f):= \{g\in G\mid f(g)=e_H\}$:
- The neutral element of $H=(\{3n\mid n\in\mathbb Z\}, +)$ is $e_H=0$. The only element of $x\in G=(\mathbb Z, +)$ for which $f(x)=3x=0$ is $x=0.$ Thus $\{0\}$ is the kernel of $f.$
Image $\operatorname{im}(f):=f[G]=\{f(g)\in H\mid g\in G\}$:
- The image $f[G]=\{3x\mid x\in\mathbb Z\}$ are all multiples of $3$, i.e. $f[G]=H.$

Kernel $\operatorname{ker}(f):= \{g\in G\mid f(g)=e_H\}$:
- The neutral element of $H=(\mathbb R^*, \cdot)$ is $e_H=1$. The only element of $x\in G=(\mathbb R,+)$ for which $\exp(x)=1$ is $x=0.$ Thus $\{0\}$ is the kernel of $\exp.$
Image $\operatorname{im}(\exp):=f[G]=\{f(g)\in H\mid g\in G\}$:
- The image $\exp[\mathbb R]=\mathbb R^+$ are all positive real numbers, i.e. $\exp[\mathbb R]=\mathbb R^+.$

Kernel $\operatorname{ker}(f):= \{g\in G\mid f(g)=e_H\}$:
- The neutral element of $H$ is $$e_H=\pmatrix{1&0\\0&1}.$$ The elements of $x\in (\mathbb R,+)$ for which $\rho(x)=e_H$ are all integer multiples of $2\pi k,$ $k\in\mathbb Z.$ This follows from special values of cosine and sine and their periodicity. Thus $\{2\pi k\mid k\in\mathbb Z\}$ is the kernel of $\rho.$
Image $\operatorname{im}(\exp):=f[G]=\{f(g)\in H\mid g\in G\}$:
- The image $\rho[\mathbb R]=\mathbb R^+$ are all matrices of $H=(\operatorname{GL}(2,\mathbb R),\cdot).$

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