Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.4 Prove that the multiplicative groups $\mathbb{R}^\times$ and $\mathbb{C}^\times$ are not isomorphic. Solution: Recall from Exercise 1.6.2…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.3 If $\varphi : G \rightarrow H$ is an isomorphism, prove that $G$ is abelian if and…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.29 Let $A$ and $B$ be groups. Prove that $A \times B$ is abelian if and only…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.25 Let $G$ be a group. Prove that if $x^2 = 1$ for all $x \in G$,…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.14 Find the orders of the following elements of the multiplicative group $(\mathbb{Z}/(36))^\times$: $\overline{1}$, $\overline{-1}$, $\overline{5}$, $\overline{13}$,…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.13 Find the orders of the following elements of the additive group $\mathbb{Z}/(36)$: $\overline{1}$, $\overline{2}$, $\overline{6}$, $\overline{9}$,…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.12 Find the orders of the following elements of the multiplicative group $(\mathbb{Z}/(12))^\times$: $\overline{1}$, $\overline{-1}$, $\overline{5}$, $\overline{7}$,…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.11 Find the orders of each element of the additive group $\mathbb{Z}/(12)$. Solution: For an element $n$…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.10 Prove that a finite group is abelian if and only if its group table is a…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.9 Let $G = \{ a+b \sqrt{2} \in \mathbb{R} \ |\ a,b \in \mathbb{Q} \}$. (1) Prove…