Hall satisfies intermediate subgroup condition

Statement
Any Hall subgroup of a finite group is also a Hall subgroup in any intermediate subgroup.

Related facts

 * Sylow satisfies intermediate subgroup condition: Essentially, the same proof.

Facts used

 * 1) uses::Index is multiplicative

Proof
Given: A finite group $$G$$, a Hall subgroup $$H$$, and a subgroup $$K$$ of $$G$$ containing $$H$$.

To prove: $$H$$ is a Hall subgroup of $$K$$.

Proof: Note that by the multiplicativity of index:

$$[G:H] = [G:K][K:H]$$.

Thus, the index $$[K:H]$$ divides the index $$[G:H]$$. In particular, if $$|H|$$ and $$[G:H]$$ are relatively prime, so are $$|H|$$ and $$[K:H]$$.