Dummit+and+foote+solutions+chapter+4+overleaf+full [repack] Access

The Sylow theorems are arguably the most important computational tool in finite group theory. For a finite group (G) with (|G| = p^n m) where (p \nmid m):

\beginproof Count pairs $(g,a)$ with $g\cdot a = a$ in two ways: $\sum_g\in G|\operatornameFix(g)| = \sum_a\in A|G_a|$. By Orbit–Stabilizer, $|G_a| = |G|/|\mathcalO_a|$. Hence \[ \sum_a\in A \fracG = |G| \sum_\textorbits O \sum_a\in O \frac1 = |G| \cdot (\text\# orbits). \] Dividing by $|G|$ gives the result. \endproof dummit+and+foote+solutions+chapter+4+overleaf+full

This is one of the most comprehensive and cleanly typeset guides available. It covers numerous chapters, including Chapter 4. You can find the unofficial solution guide on his website or via GitHub if you want to see the source code. The Sylow theorems are arguably the most important