Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.10 Let $\varphi : \mathbb{Z}/(8) \rightarrow \mathbb{Z}/(4)$ be defined by $\overline{a} \mapsto \overline{a}$. (Note that $\overline{a}$ means…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.9 Define $\varphi : \mathbb{C}^\times \rightarrow \mathbb{R}^\times$ by $$a+bi \mapsto a^2 + b^2.$$ Prove that $\varphi$ is…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.8 Let $\varphi : \mathbb{R}^\times \rightarrow \mathbb{R}^\times$ be given by $x \mapsto |x|$. Prove that $\varphi$ is…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.2 Let $\varphi : G \rightarrow H$ be a group homomorphism with kernel $K$ and let $a,b…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.1 Let $\varphi : G \rightarrow H$ be a group homomorphism and let $E \leq H$ be…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.16 Let $A$ and $B$ be groups. Prove that the maps $\pi_1 : A \times B \rightarrow…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.15 Define a map $\pi : \mathbb{R}^2 \rightarrow \mathbb{R}$ by $\pi((x,y)) = x$. Prove that $\pi$ is…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.14 Let $\varphi : G \rightarrow H$ be a group homomorphism. Define the kernel of $\varphi$ to…