The n-th powers and n-th roots of an abelian group are subgroups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.12 Let $A$ be an abelian group and fix $n \in \mathbb{Z}^+$. Prove that the following subsets…
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 $\mathbb{Z} \times \mathbb{Z}/(n)$.…
Torsion elements in an abelian group form a subgroup
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.6 Let $G$ be a group. An element $x \in G$ is called torsion if it has…
If a group has an automorphism which is fixed point free of order 2, then it is abelian
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.23 Let $G$ be a finite group which possesses an automorphism $\sigma$ such that $\sigma(g) = g$…
Power maps are abelian group homomorphisms
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.22 Let $A$ be an abelian group and fix some $k \in \mathbb{Z}$. Prove that the map…
Exhibit some automorphisms of the rational numbers
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.21 Prove that for each fixed nonzero $k \in \mathbb{Q}$ the map $\varphi : \mathbb{Q} \rightarrow \mathbb{Q}$…
The square map is a group homomorphism precisely on abelian groups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.18 Let $G$ be a group. Show that the map $\varphi : G \rightarrow G$ given by…
The inversion map is a homomorphism precisely on abelian groups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.17 Let $G$ be a group. Prove that the map $\varphi : G \rightarrow G$ given by…
The additive groups of integers and rational numbers are not isomorphic
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.6 Prove that the additive groups $\mathbb{Z}$ and $\mathbb{Q}$ are not isomorphic. Solution: First we prove a…
The additive groups of rational and real numbers are not isomorphic
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.5 Prove that the additive groups $\mathbb{Q}$ and $\mathbb{R}$ are not isomorphic. Solution: We know that no…