Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.1 Exercise 4.1.3 Suppose $G \leq S_A$ is an abelian and transitive subgroup. Show that $\sigma(a) \neq a$ for…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.9 Let $A$ be a nonempty set and let $k \in \mathbb{Z}^+$ such that $k \leq |A|$.…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.8 Let $A$ be a nonempty set and let $k \in \mathbb{Z}^+$ such that $k \leq |A|$.…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.8 Let $G = S_n$ and fix $i \in \{ 1, 2, \ldots, n \}$. Prove that…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.1 Exercise 2.1.5 Let $G$ be a finite group with $|G| = n > 2$. Prove that $G$ cannot…
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…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.6 Exercise 1.6.8 Prove that if $n \neq m$ then $S_n$ and $S_m$ are not isomorphic. Solution: We know…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.36 Assume that $G = \{1, a, b, c\}$ is a group of order 4 with identity…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.35 Let $G$ be a group, and let $x \in G$ be an element of finite order;…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.1 Exercise 1.1.34 Let $G$ be a group and $x \in G$ an element of order $n < \infty$.…