Sylow subgroups
A Sylow p-subgroup of a finite group G is a maximal p-subgroup — a
subgroup of order pⁿ, where pⁿ is the largest power of the prime p
dividing |G|. Sylow's theorems guarantee they exist, are all conjugate, and
number ≡ 1 (mod p). A normal Sylow subgroup has a complement
(Schur–Zassenhaus), splitting G as a semidirect product.