Hall satisfies intermediate subgroup condition

This article gives the statement, and possibly proof, of a subgroup property (i.e., Hall subgroup) satisfying a subgroup metaproperty (i.e., intermediate subgroup condition)
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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. 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 $[K:H]$.