Equivalence of definitions of finite solvable group

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This article gives a proof/explanation of the equivalence of multiple definitions for the term finite solvable group
View a complete list of pages giving proofs of equivalence of definitions

Statement

The following are equivalent for a finite group:

  1. It is a solvable group
  2. It is a 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 composition series for the group) are cyclic groups of prime order.