Finite solvable group

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

Definition

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

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.

Examples

Extreme examples

• The trivial group is a finite solvable group.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
finite abelian group	finite and an abelian group	follows from abelian implies solvable	see solvable not implies abelian	|FULL LIST, MORE INFO
finite nilpotent group	finite and a nilpotent group	follows from nilpotent implies solvable	see solvable not implies nilpotent	|FULL LIST, MORE INFO
finite supersolvable group	finite and a supersolvable group	follows from supersolvable implies solvable	see solvable not implies supersolvable	|FULL LIST, MORE INFO