Sylow satisfies permuting transfer condition

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 $K$.