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Compute the kernel and fibers of a given group homomorphism


Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.13

Let $G$ be the additive group of real numbers and $H$ the multiplicative group of complex numbers with absolute value 1. Let $\varphi : G \rightarrow H$ be the homomorphism $r \mapsto e^{4 \pi i r}$. Find the kernel of $\varphi$ and the fibers over $-1$, $i$, and $e^{4 \pi i / 3}$.


Lemma: Let $\alpha$, $\beta$ be set mappings such that $\alpha \circ \beta$ exists, and let $\alpha^\star$ denote preimages. Then $$(\alpha \circ \beta)^\star [A] = \beta^\star [\alpha^\star [A]].$$ Proof: If $x \in (\alpha \circ \beta)^\star [A]$, then $(\alpha \circ \beta)(x) \in A$, so that $\alpha(\beta(x)) \in A$. Thus $\beta(x) \in \alpha^\star[A]$, so that $x \in \beta^\star [\alpha^\star [A]]$. If $x \in \beta^\star [\alpha^\star [A]]$, then $\beta(x) \in \alpha^\star[A]$, so that $\alpha(\beta(x)) \in A$, hence $x \in (\alpha \circ \beta)^\star [A]$. $\square$

Note that $\varphi = \overline{\varphi} \circ \tau$, where $\overline{\varphi}$ is the mapping considered in Exercise 3.1.12 and $\tau(x) = 2x$. Using the lemma, we see that $$\mathsf{ker}\ \varphi = \{ n/2 \ |\ n \in \mathbb{Z} \},$$ $$\varphi^\star(-1) = \{ n/2 + 1/4 \ |\ n \in \mathbb{Z} \},$$ $$\varphi^\star(i) = \{ n/2 + 1/8 \ |\ n \in \mathbb{Z} \},$$ $$\varphi^\star(e^{4 \pi / 3}) = \{ n/2 + 1/3 \ |\ n \in \mathbb{Z} \}.$$

Linearity

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