Q/Z is divisible
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.15 Prove that a quotient of a divisible abelian group by any proper subgroup is also divisible.…
Q/Z is isomorphic to the group of complex roots of unity
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.14 Consider the additive quotient group $\mathbb{Q}/\mathbb{Z}$. (1) Show that every coset of $\mathbb{Z}$ in $\mathbb{Q}/\mathbb{Z}$ has…
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…
Represent the real numbers as complex numbers of modulus 1
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.12 Let $G$ be the additive group of real numbers and $H$ the multiplicative group of complex…
Exhibit a group homomorphism on the Heisenberg group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.11 Let $F$ be a field and let $$G = \left\{ \begin{bmatrix} a & b \\ 0…
Exhibit a group homomorphism from Z/(8) to Z/(4)
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.10 Let $\varphi : \mathbb{Z}/(8) \rightarrow \mathbb{Z}/(4)$ be defined by $\overline{a} \mapsto \overline{a}$. (Note that $\overline{a}$ means…
Describe the kernel and fibers of a given group homomorphism
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.9 Define $\varphi : \mathbb{C}^\times \rightarrow \mathbb{R}^\times$ by $$a+bi \mapsto a^2 + b^2.$$ Prove that $\varphi$ is…
Absolute value is a group homomorphism on the multiplicative real numbers
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…
Describe the fibers of a given group homomorphism
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.7 Define $\pi : \mathbb{R}^2 \rightarrow \mathbb{R}$ by $\pi(x,y) = x + y$. Prove that $\pi$ is…
Sign map is a homomorphism on the multiplicative group of reals
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.6 Define $\varphi : \mathbb{R}^\times \rightarrow \{ \pm 1 \}$ by $x \mapsto x / |x|$. Describe…