## Compute some orbits of an action by Sym(4) on polynomials in four variables

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.1 Exercise 4.1.6 As in Exercise 2.2.12, let $S_4$ act on the set $R$ of all polynomials with integer…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.1 Exercise 4.1.6 As in Exercise 2.2.12, let $S_4$ act on the set $R$ of all polynomials with integer…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.1 Exercise 4.1.7 Let $G \leq S_A$ act transitively on the set $A$. A block is a nonempty subset…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.1 Exercise 4.1.8 A transitive permutation group $G \leq S_A$ acting on $A$ is called doubly transitive if for…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.1 Exercise 4.1.9 Suppose $G \leq S_A$ acts transitively on $A$ and let $H \leq G$ be normal. Let…

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…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.1 Exercise 4.1.3 Suppose $G \leq S_A$ is an abelian and transitive subgroup. Show that $\sigma(a) \neq a$ for…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.1 Exercise 4.1.1 Solution: First we prove that $\mathsf{stab}_G(b) = g \mathsf{stab}_G(a) g^{-1}$. ($\subseteq$) If $x \in \mathsf{stab}_G(b)$, then…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.18 Let $H$ be a group acting on a set $A$. Prove that the relation on $A$…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.17 Let $G$ be a group and let $G$ act on itself by left conjugation; i.e., $g…

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.7 Exercise 1.7.16 Let $G$ be a group. Show that the mapping defined by $g \cdot a = gag^{-1}$…