Sylow theorems

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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 pm 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 np of Sylow p-subgroups of G is equal to [G:NG(H)], where H is any Sylow p-subgroup of G and NG(H) is the normalizer of H in G. Furthermore, np divides s, and is congruent to 1 (mod p).

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