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## 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)…

## 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…

## If a finite group has a generating set containing two elements of order 2, then it is a dihedral group

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.24 Let $G$ be a finite group and let $x$ and $y$ be distinct elements of order…

## Dih(24) and Sym(4) are not isomorphic

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.9 Prove that $D_{24}$ and $S_4$ are not isomorphic. Solution: We know from Exercise 1.2.3 that $D_{24}$…

## Dih(8) and the quaternion group are not isomorphic

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.7 Prove that $D_8$ and $Q_8$ are not isomorphic. Solution: We saw in Exercise 1.5.2 that $D_8$…

## Detecting copies of Dih(2n) in a larger group

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.2 Exercise 1.2.6 Let $G$ be a group, and let $x,y \in G$ be elements of order 2. Prove…

## 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 $S_3$:…