# Finite solvable group

From Groupprops

This article defines a property that can be evaluated for finite groups (and hence, a particular kind of group property)

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## Definition

A finite group is termed a **finite solvable group** if it satisfies the following equivalent conditions:

- 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.