Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.21 Let $p$ be an odd prime and $n$ a positive integer. Use the Binomial Theorem to…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.20 Let $p$ be a prime and $n$ a positive integer. Show that if $x$ is an…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.19 Show that if $H$ is a group and $h \in H$, there exists a unique homomorphism…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.18 We write $Z_n = \langle x \rangle$. Show that if $H$ is any group and $h…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.17 Find a presentation for $Z_n $ with one generator. Solution: We have $Z_n = \langle x…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.16 Let $G$ be a group with $x,y \in G$. Assume $|x| = n$ and $|y| =…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.15 Prove that $\mathbb{Q} \times \mathbb{Q}$ is not cyclic. Solution: Suppose $\mathbb{Q} \times \mathbb{Q}$ is cyclic. By…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.14 Let $\sigma = (1\ 2\ 3\ 4\ 5\ 6\ 7\ 8\ 9\ 10\ 11\ 12)$. For…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.13 Prove that the following pairs of groups are not isomorphic: (1) $\mathbb{Z} \times Z_2$ and $\mathbb{Z}$,…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.12 Prove that the following groups are not cyclic: (1) $Z_2 \times Z_2$, (2) $Z_2 \times \mathbb{Z}$,…