Characterize the normalizer of a cyclic subgroup
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.24 Let $G$ be a finite group and let $x \in G$. (1) Prove that if $g…
The group of units in $\mathbb Z/(2^n)$ is not cyclic for n at least 3
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.23 Show that $(\mathbb{Z}/(2^n))^\times$ is not cyclic for any $n \geq 3$. (Hint: find two distinct subgroups…
Compute the order of 5 in the integers mod a power of 2
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.22 Let $n$ be an integer with $n \geq 3$. Use the Binomial Theorem to show that…
Use the Binomial Theorem to compute the order of an element in the integers mod a prime power
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…
If a prime power power of a group element is trivial, then the order of the element is a prime power
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…
Unique group homomorphism from Z to any group where the image of 1 is fixed
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…
Group homomorphism from cyclic group is determined by the image of generator
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…
Find a group presentation for cyclic group of order n
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…
The order of a product of commuting group elements divides the least common multiple of the orders of the elements
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| =…
The direct product of two copies of the rationals is not a cyclic group
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…