The complex roots of unity form a multiplicative group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.8 Let $G = \{ z \in \mathbb{C} \ |\ z^n = 1 \mathrm{for\ some} n \in…
The reals mod the integers are an abelian group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.7 Let $G = \{ x \in \mathbb{R} \ |\ 0 \leq x < 1 \}$ and…
Determine whether or not a given set of rational numbers is additively closed
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.6 Determine which of the following sets are groups under addition: The set $A_1$ of rational numbers…
Find a presentation for the quaternion group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.5 Exercise 1.5.3 Find a set of generators and relations for $Q_8$. Solution: Let $$X = \langle x, y…
Write out the Cayley table for Sym(3), Dih(8), and the quaternion group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.5 Exercise 1.5.2 Write out the group tables for $S_3$, $D_8$, and $Q_8$. Solution: Group table for $S_3$:…
Compute the order of each element in the quaternion group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.5 Exercise 1.5.1 Compute the order of each of the elements in $Q_8$. Solution: $x$ Reasoning Order 1 1…
Z/(n) is not a group under multiplication
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.5 Prove for all $n > 1$ that $\mathbb{Z}/(n)$ is not a group under multiplication of residue…
Multiplication of residue classes of integers is associative
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.4 Prove that multiplication of residue classes in $\mathbb{Z}/(n)$ is associative. (You may assume that it is…
Addition of residue classes of integers is associative
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.3 Prove that addition of residue classes in $\mathbb{Z}/(n)$ is associative. (You may assume that it is…
Decide whether or not a given binary operator is commutative
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.2 Determine which of the following binary operations are commutative. The operation $\star$ on $\mathbb{Z}$ defined by…