Sylow theorems

The Sylow theorems of group theory form a partial converse to the theorem of Lagrange, which states that if H is a subgroup of a finite group G, then the order of H divides the order of G. The Sylow theorems guarantee, for certain divisors of the order of G, the existance of corresponding subgroups:

Let p be a prime number and write the order of G as pⁿ s (with p not dividing s). We define a Sylow p-subgroup of G to be a subgroup of G which has order pⁿ. Then:

1. If H is a subgroup of G with order p^m for some m, then H is contained in some Sylow p-subgroup of G.

2. All Sylow p-subgroups of G are conjugate to each other, i.e. if H₁ and H₂ are Sylow p-subgroups of G, then there exists an element g in G with g^-1H₁g = H₂.

3. The number n_p of Sylow p-subgroups of G is equal to [G:N_G(H)], where H is any Sylow p-subgroup of G and N_G(H) is the normalizer? of H in G. Furthermore, n_p divides s, and is congruent to 1 (mod p).

This last statement implies that G contains at least one subgroup of order pⁿ.