Solution
= Solution
A <Hall subgroup> for a prime set $\pi$ is a subgroup $H\leq G$ whose order has only prime divisors in $\pi$ and whose index has no prime divisor in $\pi$:
$$
\boxed{|H|\text{ is a }\pi\text{-number},\qquad [G:H]\text{ is a }\pi'\text{-number}.}
$$
Here one is allowed in either class, and $\pi'$ is the complementary set of primes. Equivalently $|H|$ contains the complete prime-power contribution to $|G|$ for each prime in $\pi$.