Sylow implies order-conjugate

Statement with symbols
There are two formulations:


 * Any two $$p$$-Sylow subgroups of a finite group are conjugate subgroups.
 * Suppose $$P$$ is a $$p$$-Sylow subgroup of a finite group $$G$$, and $$Q$$ is a subgroup of $$G$$ of the same order as $$G$$. Then $$P$$ and $$Q$$ are conjugate in $$G$$.

The equivalence of the formulations follows from the fact that the condition of being a $$p$$-Sylow subgroup is completely determined by the order.

Other parts of Sylow's theorem

 * Sylow subgroups exist
 * Sylow implies order-dominating: A stronger formulation of the result, that can be used to prove it.
 * Congruence condition on Sylow numbers: Another corollary of one of the proofs of this result.
 * Divisibility condition on Sylow numbers

All these facts together are known as Sylow's theorem.

Applications

 * Sylow implies isomorph-conjugate
 * Sylow implies intermediately isomorph-conjugate
 * Sylow implies automorph-conjugate
 * Sylow implies intermediately automorph-conjugate
 * Sylow implies procharacteristic
 * Sylow implies pronormal
 * Sylow of normal implies pronormal

Facts used

 * 1) uses::Sylow implies order-dominating: This states that if $$P$$ is $$p$$-Sylow and $$Q$$ is a $$p$$-subgroup, there exists a conjugate $$gPg^{-1}$$ of $$P$$ such that $$Q \le gPg^{-1}$$.
 * 2) uses::Order-dominating implies order-conjugate in finite

Proof using order-domination
This proof follows directly from facts (1) and (2).