Groupprops, The Group Properties Wiki (pre-alpha)

From Groupprops

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.

Relation with other properties

Stronger properties

• Finite Abelian group
• Finite nilpotent group
• Finite supersolvable group