Equivalence of definitions of finite solvable group

Statement
The following are equivalent for a finite group:


 * 1) It is a defining ingredient::solvable group
 * 2) It is a defining ingredient::polycyclic group
 * 3) It has Sylow complements for all prime divisors of the order of the group
 * 4) It has Hall subgroups of all possible orders
 * 5) All its composition factors (i.e., the quotient groups for any defining ingredient::composition series for the group) are cyclic groups of prime order.