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…
Transitive group actions induce transitive actions on the orbits of the action of a subgroup
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…
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…
Characterization of the orbits of a group action as equivalence classes
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 $A$…