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…
An abelian group has the same cardinality as any sets on which it acts transitively
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…
Stabilizer commutes with conjugation II
Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.1 Exercise 4.1.2 Let $G$ be a permutation group on the set $A$ (i.e. $G \leq S_A$), let $\sigma…