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…
Direct product of groups is essentially associative
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.12 Let $A$, $B$, and $C$ be groups. Show that $A \times (B \times C) \cong (A…
Direct product of groups is essentially commutative
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.11 Let $A$ and $B$ be groups. Prove that $A \times B \cong B \times A$. Solution:…
If A and B have the same cardinality, then Sym(A) and Sym(B) are isomorphic
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.10 Let $\theta : \Delta \rightarrow \Omega$ be a bijection. Define $\varphi : S_\delta \rightarrow S_\Omega$ by…