# Tagged: Dihedral Group

## 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 = \tau^2 = 1,...

## 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 of $G/H$ is 8....

## 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. Solution: We have the...

## 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 $n$ is odd. (2)...

## Demonstrate that a given subset is not a subgroup

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.2 Show that in each of the following examples, the specified subset is not a subgroup. (1) The set of 2-cycles...

## Identify Dih(2n) as a subgroup of general linear group of dimension 2 over real numbers

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.25 Let $n \in \mathbb{Z}^+$, let $r$ and $s$ be the usual generators of $D_{2n}$, and let \$\theta = 2 \pi...