Clanlu Clanlu
  • 请到 [后台->外观->菜单] 中设置菜单
  • 登录
现在登录。
  • 请到 [后台->外观->菜单] 中设置菜单

Compute the kernel and fibers of a given group homomorphism

Math 2年 前

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 $$\mathsf{ker}\ \varphi = \{ n/2 \ |\ n \in \mathbb{Z} \},$$ $$\varphi^\star(-1) = \{ n/2 + 1/4 \ |\ n \in \mathbb{Z} \},$$ $$\varphi^\star(i) = \{ n/2 + 1/8 \ |\ n \in \mathbb{Z} \},$$ $$\varphi^\star(e^{4 \pi / 3}) = \{ n/2 + 1/3 \ |\ n \in \mathbb{Z} \}.$$

#Complex Numbers#Fibers#Preimage#Root of Unity
0
Math
O(∩_∩)O哈哈~
猜你喜欢
  • The set of prime ideals of a commutative ring contains inclusion-minimal elements
  • Use Zorn’s Lemma to construct an ideal which maximally does not contain a given finitely generated ideal
  • Not every ideal is prime
  • Characterization of maximal ideals in the ring of all continuous real-valued functions
  • Definition and basic properties of the Jacobson radical of an ideal
23 5月, 2020
Compute the centralizers of each element in Sym(3), Dih(8), and the quaternion group
精选标签
  • Subgroup 40
  • Order 37
  • Counterexample 36
Copyright © 2022 Clanlu. Designed by nicetheme.