Finite supersolvable 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 supersolvable group is a finite group satisfying the following equivalent conditions:

1. It is a supersolvable group: it has a normal series where all the quotients are cyclic groups.
2. It has a chief series where all the successive quotients are groups of prime order.
3. It is a solvable group that also satisfies the property that its chief series are composition series.

Relation with other properties

Stronger properties

• Finite abelian group
• Finite nilpotent group

Weaker properties

• Group having subgroups of all orders dividing the group order: For proof of the implication, refer Finite supersolvable implies subgroups of all orders dividing the group order and for proof of its strictness (i.e. the reverse implication being false) refer Subgroups of all orders dividing the group order not implies supersolvable.
• Finite solvable group