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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…

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…

Exhibit Dihedral group as a subgroup of Symmetric group via regular representation

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.3 Solution: To save effort we will perform this computation in $S_{D_8}$ and then use the labeling…