ECD condition for pi-subgroups in solvable groups

Statement
In a finite solvable group, the set of $$\pi$$-subgroups, for any prime set $$\pi$$ (i.e., the set of subgroups that satisfy the condition that all prime divisors of the order are in $$\pi$$), satisfies the ECD condition with the maximal elements being Hall $$\pi$$-subgroups. Explicitly, for any $$\pi$$:


 * Existence (E): There exists a Hall $$\pi$$-subgroup
 * Conjugacy (C): Any two Hall $$\pi$$-subgroups are conjugate
 * Domination (D): Any $$\pi$$-subgroup is contained in a Hall $$\pi$$-subgroup. Equivalently, given any particular Hall $$\pi$$-subgroup, every $$\pi$$-subgroup is conjugate to a subgroup contained within this Hall $$\pi$$-subgroup
 * Number (N): The number of Hall $$\pi$$-subgroups is a product of factors, each of which is congruent to $$1$$ modulo some $$p \in \pi$$.

It turns out that conversely, if Hall $$\pi$$-subgroups exist for every prime set $$\pi$$ and a given finite group, then the finite group is solvable. This nontrivial result is termed Hall's theorem, and relies on the hard Burnside's p^aq^b theorem, which proves it for the case where there are only two prime divisors of the order.

Proof components

 * 1) uses::Hall subgroups exist in finite solvable
 * 2) uses::Hall implies order-conjugate in finite solvable
 * 3) uses::Hall implies order-dominating in finite solvable
 * 4) uses::Congruence condition on factorization of Hall numbers

Also related is: Divisibility condition on factorization of Hall numbers.