I am working on problem 4.1.9 in Dummit and Foote's Abstract Algebra. Thought I am in an algebra course, we don't use this book, and this isn't a homework assignment.
G acts transitively on a finite set A and H is a normal subgroup of G. are the distinct orbits of H on A.
I have already done half of the problem, which is to show all orbits of H have the same cardinality and that G is transitive on the set where G acts on this set via representative.
My problem is this: Prove that if then and prove that .
I cannot do the second part (the first isn't hard). A hint is to use the 2nd isomorphism theorem, but I don't see how it helps.