Hall is transitive

This article gives the statement, and possibly proof, of a subgroup property (i.e., Hall subgroup) satisfying a subgroup metaproperty (i.e., transitive subgroup property)
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Statement

Verbal statement

Any Hall subgroup of a Hall subgroup of a finite group, is a Hall subgroup in the whole group.

Facts used

1. Index is multiplicative
2. Lagrange's theorem

Proof

Given: A finite group $G$, subgroups $H\leq K\leq G$ such that $H$ is a Hall subgroup of $K$ and $K$ is a Hall subgroup of $G$.

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

Proof: By fact (1), we have:

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

Now, since $H$ is Hall in $K$, the order of $H$ is relatively prime to $[K:H]$.

By fact (2), the order of $H$ divides the order of $K$, and since $K$ is a Hall subgroup of $G$ ,the order of $K$ is relatively prime to $[G:K]$. Thus, the order of $H$ is relatively prime to $[G:K]$.

Thus, the order of $H$ is relatively prime to the product $[G:K][K:H]$, which, by the above equation, equals $[G:H]$.