## Use Sylow’s Theorem to prove Cauchy’s Theorem

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.5 Exercise 4.5.1 Solution: Let $G$ be a finite group and $p$ a prime dividing $|G|$. By Sylow’s Theorem,…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.5 Exercise 4.5.1 Solution: Let $G$ be a finite group and $p$ a prime dividing $|G|$. By Sylow’s Theorem,…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.13 Solution: $G$ contains an element $x$ of order 2 by Cauchy’s Theorem. Let $\pi : G…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.4 Exercise 3.4.1 Solution: Let $G$ be an abelian simple group. Suppose $G$ is infinite. If $x \in G$…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.9 Solution: (1) We prove this equality by attempting to choose an arbitrary element of $\mathcal{S}$. Note…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.1 Which of the following are permissible orders for subgroups of a group of order 120: 1,…