Compute in a group ring over dihedral group of order 8
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.2 Exercise 7.2.9 Let $\alpha = r + r^2 - 2s$ and $\beta = -3r^2 + rs$ be elements…
Draw subgroup lattice of a quotient of quasi-dihedral group of order 16
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.3 Exercise 3.3.5 Let $QD_{16} = \langle \sigma, \tau \rangle$ be the quasidihedral group of order 16 described in…
Perform explicit computation in a quotient of a quasi-dihedral group
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…
Perform computations in a quotient of dihedral group of order 16
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.1 Exercise 3.1.17 Let $G = D_{16}$ and let $H = \langle r^4 \rangle$. (1) Show that the order…
Exhibit the cyclic subgroups of Dih(8) as sets
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.11 Find all cyclic subgroups of $D_8$. Exhibit a proper subgroup of $D_8$ which is not cyclic.…
Compute the center of Dih(2n)
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.7 Let $n \in \mathbb{Z}$ with $n \geq 3$. Prove the following. (1) $Z(D_{2n}) = 1$ if…
Demonstrate that a given subgroup is normal
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.5 For each of the following subgroups $A$ of a given group $G$, show that $C_G(A) =…
Compute the centralizers of each element in Sym(3), Dih(8), and the quaternion group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.4 For each of the groups $S_3$, $D_8$, and $Q_8$, compute the centralizer of each element and…
Set of all elements of a given order is not necessarily a subgroup
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.14 Let $n \geq 3$. Show that $\{ x \in D_{2n} \ |\ x^2 = 1\}$ is…
Demonstrate that given subsets of Dih(8) are subgroups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.3 Show that the following subsets of $D_8$ are subgroups. (1) $\{ 1, r^2, s, sr^2 \}$…