Chapter 4 of Abstract Algebra by David S. Dummit and Richard M. Foote is titled Group Actions
Most Sylow problems are "counting games." Use the congruence and the fact that must divide the index to narrow down the possibilities. dummit foote solutions chapter 4
If you have a specific problem from Chapter 4 you're struggling with, please provide the problem number or describe it, and I'll do my best to guide you through it step-by-step. Chapter 4 of Abstract Algebra by David S