Exhibit two Sylow 2-subgroups of Symmetric group $S_5$
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.5 Exercise 4.5.8 Solution: $|S_5| = 120 = 2^3 \cdot 3 \cdot 5$, so that the Sylow 2-subgroups of…
Exhibit all of the Sylow 2-subgroups of Symmetric group $S_4$
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.5 Exercise 4.5.7 Solution: The Sylow 2-subgroups of $S_4$ have order 8, and by Sylow’s Theorem, $n_2 = 1…
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…
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…
Interaction of Sylow subgroups with quotients
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 3.3 Exercise 3.3.9 Let $p$ be a prime and let $G$ be a finite group of order $p^am$, where…