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| =…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.3 Solution: Let $x = (x_j)$ and $y = (y_j)$. If $j \in I$, then $$(xy)_j =…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.4 Exercise 3.4.1 Solution: Let $G$ be an abelian simple group. Suppose $G$ is infinite. If $x \in G$…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.37 Solution: First we prove a lemma. Lemma: Let $R$ be a ring, and let $I_1, I_2,…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.36 Solution: We begin with a lemma. Lemma: If $R$ is a commutative ring with 1 and…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.35 Solution: (1) We show that $I(J+K) = IJ + IK$; the proof of the other equality…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.34 Solution: (1) We wish to prove the following: (a) $I+J$ is an ideal of $R$ and…