# Tagged: Counterexample

## The order of a product of commuting group elements divides the least common multiple of the orders of the elements

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.16 Let $G$ be a group with $x,y \in G$. Assume $|x| = n$ and $|y| = m$. Suppose that $x$...

## The direct product of two copies of the rationals is not a cyclic group

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.15 Prove that $\mathbb{Q} \times \mathbb{Q}$ is not cyclic. Solution: Suppose $\mathbb{Q} \times \mathbb{Q}$ is cyclic. By Theorem 7 in the...

## Prove that two given groups are nonisomorphic

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.13 Prove that the following pairs of groups are not isomorphic: (1) $\mathbb{Z} \times Z_2$ and $\mathbb{Z}$, (2) $\mathbb{Q} \times Z_2$...

## If n is composite, then Z/(n) is not a field

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.4 Exercise 1.4.4 Show that if $n$ is not prime, then $\mathbb{Z}/(n)$ is not a field. Solution: If $n$ is not prime, then...

## Show that a given general linear group is nonabelian

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.4 Exercise 1.4.3 Show that $GL_2(\mathbb{F}_2)$ is non-abelian. Solution: We have \left[ {1 \atop 1}{1 \atop 0} \right] \cdot \left[ {0 \atop 1}{1...

## Every subgroup is contained in its normalizer

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.6 Let $G$ be a group and $H \leq G$. (1) Show that $H \leq N_G(H)$. Give an example to show...

## Compute a torsion subgroup

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.7 Fix $n \in \mathbb{Z}^+$ with $n > 1$. Find the torsion subgroup of $\mathbb{Z} \times \mathbb{Z}/(n)$. Show that the set...