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…
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,…
The Sylow numbers of a direct product are products of the Sylow numbers
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.4 Solution: Let $A$ and $B$ be finite groups and $p$ a prime, and write $|A| =…