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## Example: Examples of Group Homomorphisms

### Example 1

$f:\mathbb Z\to\mathbb Z$ with $a\to 3a$ is a group homomorphism between the group $(\mathbb Z,+)$ and its subgroup $(\{3n\mid n\in\mathbb Z\}, +),$ since $$f(a+b)=3(a+b)=3a+3b=f(a)+f(b)$$ for all integers $a,b\in\mathbb Z.$

### Example 2

For real numbers $x\in\mathbb R$, the exponential function is a group homomorphism between the groups $(\mathbb R,+)$ and $(\mathbb R^{ +},\cdot),$ since $$\exp(x+y)=\exp(x)\cdot\exp(y)$$ for all $x,y\in\mathbb R.$ This is also known as the functional equation of the exponential function.

### Example 3

The rotation matrix map $$\rho:x\to\pmatrix{\cos(x)&-\sin(x)\\\sin(x)&\cos(x)}$$ is group homomorphism of the group $(\mathbb R,+)$ and the general linear group $(\operatorname{GL}(2,\mathbb R),\cdot)$ together with the matrix multiplication $”\cdot”$. This is because of the additivity theorems for cosine and sine \begin{align}\rho(x+y)&=\pmatrix{\cos(x+y)&-\sin(x+y)\\\sin(x+y)&\cos(x+y)}\nonumber\\ &=\pmatrix{\cos(x)\cos(y)-\sin(x)\sin(y)&-\sin(x)\cos(y)-\cos(x)\sin(y)\\\sin(x)\cos(y)+\cos(x)\sin(y)&\cos(x)\cos(y)-\sin(x)\sin(y)}\nonumber\\ &=\pmatrix{\cos(x)&-\sin(x)\\\sin(x)&\cos(x)}\cdot\pmatrix{\cos(y)&-\sin(y)\\\sin(y)&\cos(y)}\nonumber\\ &=\rho(x)\cdot\rho(y).\nonumber\end{align}