Group homomorphism from cyclic group is determined by the image of generator
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.18 We write $Z_n = \langle x \rangle$. Show that if $H$ is any group and $h…
When is a map from Z/(48) to Cyc(36) a group homomorphism?
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.9 Let $Z_{36} = \langle x \rangle$. For which integers $a$ does the map $\psi_a$ defined by…
Identify quaternion group as a subgroup of the general linear group of dimension 2 over complex numbers
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.26 Let $i$ and $j$ be the generators of $Q_8 = \langle i, j \ |\ i^4…
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…
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…
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…