Exhibit quaternion group in Symmetric group via regular representation
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.4 Solution: Recall that $Q_8 = \langle i,j \rangle$. Now $i(1) = i$, $i(-1) = -i$, $i(i)…
Exhibit Dihedral group as a subgroup of Symmetric group via regular representation
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.3 Solution: To save effort we will perform this computation in $S_{D_8}$ and then use the labeling…
Exhibit Sym(3) as a subgroup of Sym(6) via the left regular representation
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.2 Solution: We use the notation $\sigma(k) = \sigma \cdot k$. (1) $1 \mapsto 1$ (2) We…
Exhibit the Klein 4-group as a subgroup of Sym(4) using the left regular representation
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.1 Solution: The multiplication table for $G$ is as follows. 1 a b c 1 1…
Use direct products to construct groups with some given properties
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.18 Solution: (1) Consider $\prod_{\mathbb{N}} \mathbb{Z}/(2)$; clearly this group is infinite, and moreover $$2(\prod x_i) = \prod…
Basic properties of the direct sum of groups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.17 Solution: (1) Note that $\prod 1 \in H$, where we may take $J = \emptyset$. Thus…
Basic properties of n-ary direct products of groups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.16 Solution: (1) First we show that $\iota_k$ is an injective homomorphism. If $g,h \in G_k$, then…
Definition of the n-ary direct product of groups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.15 Solution: (1) Elements in a direct product are uniquely represented, so well-definedness is not an issue.…
A quotient by a product is isomorphic to the product of quotients
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.14 Solution: We begin with some lemmas. Lemma 1: Let $\varphi_1 : G_1 \rightarrow H_1$ and $\varphi_2…
Compute presentations for a given central product of groups
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.13 Solution: Note that $Z_4 \times D_8$ is generated by $(x,1)$, $(1,r)$, and $(1,s)$. Hence $Z_4 \ast_\varphi…