Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.12 Solution: If $\pi[G] \leq S_G$ contains an odd permutation, then $\pi[G] \not\leq A_G$. Now $A_G \vartriangleleft…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.11 Solution: This action of $G$ is faithful, so that the induced action of $H = \langle…
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)…
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…
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…
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…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.15 Let $G$ be any group. Show that the mapping defined by $g \cdot a = ag^{-1}$…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.14 Let $G$ be a nonabelian group. Show that the action of $G$ on itself by $g…
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.13 Find the kernel of the left regular action. Solution: Recall that the left regular action of…