Linearity, Author at Solutions to Linear Algebra Done Right

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…

Compute all Sylow subgroups of Dihedral group $D_{12}$ and $S_3\times S_3$

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.5 Exercise 4.5.1 Solution: The Sylow 2-subgroups of $D_{12}$ have order 4. By Sylow’s Theorem, the number $n_2$ of…

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,…

In a finite group, conjugation permutes sub-Sylow subgroups

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.5 Exercise 4.5.2 Solution: Note that $gQg^{-1} \leq gHg^{-1}$. Moreover, because conjugation by g is an automorphism, $|gQg^{-1}| =…

Sylow subgroups of a finite group which are contained in some other subgroup are also Sylow there

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.5 Exercise 4.5.1 Solution: If $P \leq H \leq G$ is a Sylow $p$-subgroup of $G$, then $p$ does…

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 $G$…

If a group has order 2k where k is odd, then it has a subgroup of index 2

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…

If regular representation of a group G contains an odd permutation, then G has a subgroup of index 2

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.12 Solution: If $\pi[G] \leq S_G$ contains an odd permutation, then $\pi[G] \not\leq A_G$. Now $A_G \vartriangleleft…

Characterization of parity in the left regular representation of a finite group

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.11 Solution: This action of $G$ is faithful, so that the induced action of $H = \langle…

Classify groups of order 6

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.10 Solution: Let $G$ be a nonabelian group of order 6. We claim that if $x$ is…