The Sylow Theorems

Abstract Algebra · Axiom Academy

Three powerful theorems that reveal the structure of finite groups through their p-subgroups, using the elegant machinery of group actions. 1. The Setup: Prime Power Subgroups The Sylow theorems answer three fundamental questions about these maximal p-power subgroups: Do they exist? How are they related? How many are there? 2. First Sylow Theorem: Existence Proof Idea: We use a counting argument with group actions. Consider the set X of all subsets of G with p k elements. Then |X| = C(|G|, p k ). We can show this binomial coefficient is not divisible by p, so when G acts on X by left multiplication, not all orbits have size divisible by p. An orbit of size not divisible by p gives us our subgroup! 3. Second Sylow Theorem: Conjugacy Proof Idea: Let P be a Sylow p-subgroup and let Q be any p-subgroup. Consider Q acting on the set of left cosets G/P by left multiplication. The orbits partition G/P, and since |G/P| is coprime to p, at least one orbit has size coprime to p. This orbit has size 1, meaning there's a fixed point gP where Q stabilizes it. This forces Q ≤ gPg⁻¹. If Q is also Sylow, they're equal! 4. Third Sylow Theorem: Counting Proof Idea: Let S be the set of all Sylow p-subgroups. Fix one Sylow p-subgroup P. Then G acts on S by conjugation (from the second theorem, this action is transitive). The orbit-stabilizer theorem gives |S| = [G : N G (P)], where N G (P) is the normalizer. Since P ≤ N G (P), we know p k | |N G (P)|, so n p = [G : N G (P)] divides m.

This is the written version of the interactive lesson above. See the full Abstract Algebra course.