Sylow satisfies permuting transfer condition

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 $$K$$.

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) uses::Index is multiplicative
 * 2) uses::Lagrange's theorem
 * 3) uses::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 $$K$$.