Sylow Theorems - Proof of The Sylow Theorems

Proof of The Sylow Theorems

The Sylow theorems have been proved in a number of ways, and the history of the proofs themselves are the subject of many papers including (Waterhouse 1980), (Scharlau 1988), (Casadio & Zappa 1990), (Gow 1994), and to some extent (Meo 2004).

One proof of the Sylow theorems exploit the notion of group action in various creative ways. The group G acts on itself or on the set of its p-subgroups in various ways, and each such action can be exploited to prove one of the Sylow theorems. The following proofs are based on combinatorial arguments of (Wielandt 1959). In the following, we use a | b as notation for "a divides b" and a b for the negation of this statement.

Theorem 1: A finite group G whose order |G| is divisible by a prime power pk has a subgroup of order pk.

Proof: Let |G| = pkm = pk+ru such that p does not divide u, and let Ω denote the set of subsets of G of size pk. G acts on Ω by left multiplication. The orbits Gω = {gω | g ∈ G} of the ω ∈ Ω are the equivalence classes under the action of G.

For any ω ∈ Ω consider its stabilizer subgroup G_ω. For any fixed element α ∈ ω the function maps G_ω to ω injectively: for any two g, h ∈ G_ω we have that gα = hα implies g = h, because α ∈ ω ⊆ G means that one may cancel on the right. Therefore pk = |ω| ≥ |G_ω|.

On the other hand

and no power of p remains in any of the factors inside the product on the right. Hence ν_p(|Ω|) = ν_p(m) = r. Let R ⊆ Ω be a complete representation of all the equivalence classes under the action of G. Then,

Thus, there exists an element ω ∈ R such that s := ν_p(|Gω|) ≤ ν_p(|Ω|) = r. Hence |Gω| = psv where p does not divide v. By the stabilizer-orbit-theorem we have |G_ω| = |G| / |Gω| = pk+r-su/v. Therefore pk | |G_ω|, so pk ≤ |G_ω| and G_ω is the desired subgroup.

Lemma: Let G be a finite p-group, let G act on a finite set Ω, and let Ω₀ denote the set of points of Ω that are fixed under the action of G. Then |Ω| ≡ |Ω₀| mod p.

Proof: Write Ω as a disjoint sum of its orbits under G. Any element x ∈ Ω not fixed by G will lie in an orbit of order |G|/|G_x| (where G_x denotes the stabilizer), which is a multiple of p by assumption. The result follows immediately.

Theorem 2: If H is a p-subgroup of G and P is a Sylow p-subgroup of G, then there exists an element g in G such that g−1Hg ≤ P. In particular, all Sylow p-subgroups of G are conjugate to each other (and therefore isomorphic), i.e. if H and K are Sylow p-subgroups of G, then there exists an element g in G with g−1Hg = K.

Proof: Let Ω be the set of left cosets of P in G and let H act on Ω by left multiplication. Applying the Lemma to H on Ω, we see that |Ω₀| ≡ |Ω| = mod p. Now p by definition so p |Ω₀|, hence in particular |Ω₀| ≠ 0 so there exists some gP ∈ Ω₀. It follows that for some g ∈ G and ∀ h ∈ H we have hgP = gP so g−1HgP = P and therefore g−1Hg ≤ P. Now if H is a Sylow p-subgroup, |H| = |P| = |gPg−1| so that H = gPg−1 for some g ∈ G.

Theorem 3: Let q denote the order of any Sylow p-subgroup of a finite group G. Then n_p | |G|/q and n_p ≡ 1 mod p.

Proof: By Theorem 2, n_p =, where P is any such subgroup, and N_G(P) denotes the normalizer of P in G, so this number is a divisor of |G|/q. Let Ω be the set of all Sylow p-subgroups of G, and let P act on Ω by conjugation. Let Q ∈ Ω₀ and observe that then Q = xQx−1 for all x ∈ P so that P ≤ N_G(Q). By Theorem 2, P and Q are conjugate in N_G(Q) in particular, and Q is normal in N_G(Q), so then P = Q. It follows that Ω₀ = {P} so that, by the Lemma, |Ω| ≡ |Ω₀| = 1 mod p.