The group of complex p-power roots of unity is a proper quotient of itself
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.3 Exercise 3.3.8 Let $p$ be a prime and let $G$ be the group of $p-$power roots of 1 in…
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…