Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.5 Exercise 4.5.1 Solution: The Sylow 2-subgroups of $D_{12}$ have order 4. By Sylow’s Theorem, the number $n_2$ of…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.18 Solution: (1) Consider $\prod_{\mathbb{N}} \mathbb{Z}/(2)$; clearly this group is infinite, and moreover $$2(\prod x_i) = \prod…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.16 Solution: (1) First we show that $\iota_k$ is an injective homomorphism. If $g,h \in G_k$, then…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.15 Solution: (1) Elements in a direct product are uniquely represented, so well-definedness is not an issue.…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.14 Solution: We begin with some lemmas. Lemma 1: Let $\varphi_1 : G_1 \rightarrow H_1$ and $\varphi_2…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.8 Solution: First we show that this mapping $\psi : \pi \mapsto \varphi_\pi$ is a homomorphism. Let…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.7 Solution: We need to show that $\varphi_\pi$ is a bijective homomorphism. Homomorphism: Let $g = (g_i)$…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.6 Solution: Let $H \leq Q_8 \times E_{2^k}$ be a subgroup, and let $\alpha = (a,x) \in…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.5 Solution: Write $Z_4 = \langle x \rangle$ and consider $\langle (i,x) \rangle = \{ (i,x), (-1,x^2),…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.4 Solution: Let $A$ and $B$ be finite groups and $p$ a prime, and write $|A| =…