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 $\lambda : G \rightarrow S_{G/H}$…
The quaternion group is not a subgroup of Symmetric group for any n less than 8
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.7 Solution: (1) $Q_8$ is a subgroup of $S_8$ via the left regular representation. (2) Now suppose…
Compute a permutation representation of Dihedral group of order 8
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.6 Solution: We compute the permutation representation of $G$ in $D_{G/N}$.$$1 \mapsto 1$$ $$r(N) = rN, r(rN)…
Compute permutation representations of Dihedral group of order 8
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.5 Solution: To save effort, we compute the representation of $D_8$ in $S_{D_8}$.$$1 \mapsto 1$$ $$r(H) =…
Exhibit quaternion group in Symmetric group via regular representation
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.4 Solution: Recall that $Q_8 = \langle i,j \rangle$. Now $i(1) = i$, $i(-1) = -i$, $i(i)…
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…
Exhibit Sym(3) as a subgroup of Sym(6) via the left regular representation
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.2 Solution: We use the notation $\sigma(k) = \sigma \cdot k$. (1) $1 \mapsto 1$ (2) We…
Exhibit the Klein 4-group as a subgroup of Sym(4) using the left regular representation
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.1 Solution: The multiplication table for $G$ is as follows. 1 a b c 1 1…
Use direct products to construct groups with some given properties
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.18 Solution: (1) Consider $\prod_{\mathbb{N}} \mathbb{Z}/(2)$; clearly this group is infinite, and moreover $$2(\prod x_i) = \prod…