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 set of all group automorphisms of a fixed group is a group
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.20 Let $G$ be a group and let $\mathsf{Aut}(G)$ be the set of all isomorphisms $G \rightarrow…
Surjective group endomorphisms need not be automorphisms
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.19 Let $G = \{ z \in \mathbb{C} \ |\ z^n = 1\ \mathrm{for\ some}\ n \in…
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 coordinate projections from a direct product of groups are group homomorphisms
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.16 Let $A$ and $B$ be groups. Prove that the maps $\pi_1 : A \times B \rightarrow…
Compute the kernel of a coordinate projection
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.15 Define a map $\pi : \mathbb{R}^2 \rightarrow \mathbb{R}$ by $\pi((x,y)) = x$. Prove that $\pi$ is…
The kernel of a group homomorphism is a subgroup
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.14 Let $\varphi : G \rightarrow H$ be a group homomorphism. Define the kernel of $\varphi$ to…
The image of a group homomorphism is a subgroup
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.13 Let $G$ and $H$ be groups and let $\varphi : G \rightarrow H$ be a group…