# Equivalence of definitions of finite solvable group

From Groupprops

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:

- It is a solvable group
- It is a polycyclic group
- It has Sylow complements for all prime divisors of the order of the group
- It has Hall subgroups of all possible orders
- All its composition factors (i.e., the quotient groups for any composition series for the group) are cyclic groups of prime order.