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The number of Sylow p-subgroups divides the p'-part of |G| and is ≡ 1 mod p
Sylow's Third Theorem. Let be a finite group with where is prime and . Let denote the number of Sylow p-subgroups of . Then:
This theorem is a counting consequence of Sylow's second theorem together with the normalizer and the conjugation action on the set of Sylow -subgroups.
A group is a set together with a binary operation such that:
Equivalently, a group is a monoid in which every element is invertible. Much of group theory studies subgroups and structure-preserving maps called group homomorphisms.
Sylow's Second Theorem. Let be a finite group, let be a prime, and let be a Sylow p-subgroup of . If is any p-group (equivalently, is a power of ), then there exists such that
In particular, any two Sylow -subgroups of are conjugate (they lie in the same orbit under the conjugation action).
Let be a group. The conjugation action of on itself is the group action