## Perform explicit computation in a quotient of the modular group of order 16

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.19 Let $G = M = \langle u,v \ |\ u^2 = v^8 = 1, vu =…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.19 Let $G = M = \langle u,v \ |\ u^2 = v^8 = 1, vu =…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.18 Let $G = QD_{16}$ be the quasidihedral group presented by $$\langle \sigma, \tau \ |\ \sigma^8…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.5 Let $G$ be a group and $N$ a normal subgroup of $G$. Prove that the order…

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…

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 2.3.16 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 2.3.10 What is the order of $\overline{30}$ in $\mathbb{Z}/(54)$? Write out all of the elements and their…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.2 Let $G$ be a finite group and let $x \in G$. Prove that if $|x| =…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.4 Exercise 1.4.6 Let $F$ be a field. If $|F| = q$ is finite show that $|GL_n(F)| < q^{n^2}$.…