Sylow implies order-conjugate
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., Sylow subgroup) must also satisfy the second subgroup property (i.e., order-conjugate subgroup)
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Statement with symbols
There are two formulations:
- Any two -Sylow subgroups of a finite group are conjugate subgroups.
- Suppose is a -Sylow subgroup of a finite group , and is a subgroup of of the same order as . Then and are conjugate in .
The equivalence of the formulations follows from the fact that the condition of being a -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.
- Sylow implies order-dominating: This states that if is -Sylow and is a -subgroup, there exists a conjugate of such that .
- Order-dominating implies order-conjugate in finite
Proof using order-domination
This proof follows directly from facts (1) and (2).