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## For odd primes $p$, the Sylow $p$-subgroups of Diherdral group are cyclic and normal

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.5 Exercise 4.5.5 Solution: Let $P \leq D_{2n}$ be a Sylow $p$-subgroup, where $p$ is an odd prime. Note…

## A finite group of composite order n having a subgroup of every order dividing n is not simple

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.14 Solution: Let $p$ be the smallest prime dividing $n$, and write $n = pm$. Now $G$…

## In a p-group, every proper subgroup of minimal index is normal

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.9 Solution: Let $G$ be a group of order $p^k$ and $H \leq G$ a subgroup with…

## Subgroups of finite index force the existence of normal subgroups of bounded index

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.8 Solution: $G$ acts on the cosets $G/H$ by left multiplication. Let $\lambda : G \rightarrow S_{G/H}$…

## Transitive group actions induce transitive actions on the orbits of the action of a subgroup

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.1 Exercise 4.1.9 Suppose $G \leq S_A$ acts transitively on $A$ and let $H \leq G$ be normal. Let…

## The intersection by an abelian normal subgroup is normal in the product

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.20 Let $G$ be a group and $A,B \leq G$ be subgroups such that $A$ is abelian…

## Normal subgroups whose order and index are coprime are unique up to order

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.19 Let $G$ be a finite group, $N \leq G$ a normal subgroup, and suppose that $|N|$…

## If the index and order of a normal subgroup and subgroup are relatively prime, then the subgroup is contained in the normal subgroup

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.2 Exercise 3.2.18 Let $G$ be a group and let $H,N \leq G$ with $N$ normal in $G$. Prove…