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…
Decide whether or not a given binary operator is associative
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.1 Determine which of the following binary operations are associative. The operation $\star$ on $\mathbb{Z}$ defined by…
Demonstrate that a given relation defined in terms of a surjective function is an equivalence relation
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 0.1 Exercise 0.1.7 Let $f : A \rightarrow B$ be a surjective map of sets. Prove that the relation…
Determine whether a given relation is well defined
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 0.1 Exercise 0.1.6 Determine whether the function $f : \mathbb{R}^+ \rightarrow \mathbb{Z}$ given by mapping a real number $r$…