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
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.
- The trivial group is a finite solvable group.
Relation with other 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||Finite nilpotent group, Finite supersolvable group, Group having subgroups of all orders dividing the group order|FULL LIST, MORE INFO|
|finite nilpotent group||finite and a nilpotent group||follows from nilpotent implies solvable||see solvable not implies nilpotent||Finite supersolvable group, Group having a Sylow tower, Group having subgroups of all orders dividing the group order|FULL LIST, MORE INFO|
|finite supersolvable group||finite and a supersolvable group||follows from supersolvable implies solvable||see solvable not implies supersolvable||Group having subgroups of all orders dividing the group order|FULL LIST, MORE INFO|