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…
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) =…
Symmetric group acts on the n-fold direct product of a group by permuting the factors
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.8 Solution: First we show that this mapping $\psi : \pi \mapsto \varphi_\pi$ is a homomorphism. Let…
Finite direct products are isomorphic up to permutation of the factors
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 5.1 Exercise 5.1.7 Solution: We need to show that $\varphi_\pi$ is a bijective homomorphism. Homomorphism: Let $g = (g_i)$…
Compute the orbits, cycle decompositions, and stabilizers of some given group actions of Sym(3)
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.1 Exercise 4.1.4 Let $S_3$ act on the set $A = \{ (i,j) \ |\ 1 \leq i,j \leq…
A group action is faithful precisely when the kernel of the corresponding permutation representation is trivial
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.6 Prove that a group $G$ acts faithfully on a set $A$ if and only if the…
The kernel of a group action is precisely the kernel of the induced permutation representation
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.5 Prove that the kernel of an action of the group $G$ on a set $A$ is…
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