Sylow satisfies permuting transfer condition

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This article gives the statement, and possibly proof, of a subgroup property (i.e., Sylow subgroup) satisfying a subgroup metaproperty (i.e., permuting transfer condition)
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Statement

Statement with symbols

Suppose H is a Sylow subgroup of a finite group G. Suppose, further, that K is a subgroup of G such that H and K are permuting subgroups -- in other words, HK = KH. Then, H \cap K is a Sylow subgroup of K.

Definitions used

Sylow subgroup

Further information: Sylow subgroup

A subgroup H of a finite group G is termed a Sylow subgroup if its order |H| and its index [G:H] are relatively prime.

Facts used

  1. Index is multiplicative
  2. Lagrange's theorem
  3. Product formula: This states that if H and K are subgroups of G, we have:

|HK| = \frac{|H||K|}{|H \cap K|}.

Proof

Given: A finite group G, a Sylow subgroup H of G, a subgroup K of G such that HK = KH.

To prove: H \cap K is Hall in K.

Proof: Rearranging the product formula (fact (3)) yields:

\frac{|K|}{|H \cap K|} = \frac{|HK|}{|H|}.

By Lagrange's theorem (fact (2)), and noting that HK is a subgroup of G, we get:

[K:H \cap K] = [HK:H].

By fact (1), we have:

[HK:H][G:HK] = [G:H].

Thus, we get:

[K:H \cap K][G:HK] = [G:H].

In particular, [K:H \cap K] divides [G:H]. By Lagrange's theorem, we have that |H \cap K| divides |H|. Since |H| and [G:H] are relatively prime, we obtain that [K:H \cap K] and |H \cap K| are relatively prime. Thus, H \cap K is a Sylow subgroup of K.