Sylow Theorems

Sylow Theorems

Let G be a finite group.

Theorem 1

For every prime factor p with multiplicity n of the order of G, there exists a Sylow p-subgroup of G, of order p^n.

Theorem 2

All Sylow p-subgroups of G are conjugate to each other, i.e.\ if H and K are Sylow p-subgroups of G, then there exists an element g\in G with g^{-1}Hg=K.

Theorem 3

Let p be a prime such that |G|=p^nm, where p\nmid m. Let n_p be the number of Sylow p-subgroups of G. Then:
1) n_p\mid m, which is the index of the Sylow p-subgroup in G.
2) n_p\equiv 1\pmod p.

Theorem 3b (Proof)

We have n_p=[G:N_G(P)], where P is any Sylow p-subgroup of G and N_G denotes the normalizer.

Proof

Let P be a Sylow p-subgroup of G and let G act on \text{Syl}_p(G) by conjugation. We have |\text{Orb}(P)|=n_p, \text{Stab}(P)=\{g\in G:gPg^{-1}=P\}=N_G(P).

By the Orbit-Stabilizer Theorem, |\text{Orb}(P)|=[G:\text{Stab}(P)], thus n_p=[G:N_G(P)].

Orbit-Stabilizer Theorem

Let G be a group which acts on a finite set X. Then \displaystyle |\text{Orb}(x)|=[G:\text{Stab}(x)]=\frac{|G|}{|\text{Stab}(x)|}.

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