Sylow implies order-dominating

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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a finite group. That is, it states that in a Finite group (?), every subgroup satisfying the first subgroup property (i.e., Sylow subgroup (?)) must also satisfy the second subgroup property (i.e., Order-dominating subgroup (?)). In other words, every Sylow subgroup of finite group is a order-dominating subgroup of finite group.
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Statement

Property-theoretic statement

The subgroup property of being a Sylow subgroup is stronger than, or implies, the subgroup property of being an order-dominating subgroup: for any subgroup whose order divides the order of the Sylow subgroup, some conjugate of that subgroup is contained in the given Sylow subgroup.

Statement with symbols

Suppose G is a finite group and p is a prime number. Then, if P is a p-Sylow subgroup and Q is a p-subgroup of G, there exists g \in G such that g^{-1}Qg \le P.

This statement is a part of Sylow's theorem.

Related facts

Corollaries

Proof

Proof using coset spaces

Given: G a finite group, P a p-Sylow subgroup, and Q a p-group. Suppose n=p^rm where n is the order of G and m is relatively prime to p (So, |P| = p^r.

To prove: There exists g \in G such that g^{-1}Qg \le P.

Proof: We prove this through a series of observations:

  • G naturally acts (on the left) on the left coset space of P.
  • Since Q is a subgroup of G, Q also acts on the left coset space of P
  • The left coset space of P has cardinality m, which is relatively prime to p. Hence, under the action of Q (which is a p-group) on this set, there is at least one fixed point. Let the fixed point be gP.
  • We have QgP = gP. This gives us: (g^{-1}Qg)P = P, and hence g^{-1}Qg \subseteq P. Thus, a conjugate of Q is contained inside P.