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## 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…

## Basic properties of blocks of a group action

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…

## Every doubly transitive group action is primitive

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…

## 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…

## Conjugation is a group action

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 $g \cdot a = gag^{-1}$…