Hall is transitive

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

Facts used

 * 1) uses::Index is multiplicative
 * 2) uses::Lagrange's theorem

Proof
Given: A finite group $$G$$, subgroups $$H \le K \le G$$ 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 $$[G:H]$$.