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:
- It is a supersolvable group: it has a normal series where all the quotients are cyclic groups.
- It has a chief series where all the successive quotients are groups of prime order.
- It is a solvable group that also satisfies the property that its chief series are composition series.
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: any two elements commute | symmetric group:S3 is a counterexample | |FULL LIST, MORE INFO | |
| finite nilpotent group | finite and a nilpotent group | symmetric group:S3 is a counterexample | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| group having subgroups of all orders dividing the group order | for every natural number dividing the order, there is a subgroup with that natural number as order | finite supersolvable implies subgroups of all orders dividing the group order | subgroups of all orders dividing the group order not implies supersolvable | |FULL LIST, MORE INFO |
| finite solvable group | finite and a solvable group. This only requires a chief series with abelian quotients, or a composition series with cyclic quotients | |FULL LIST, MORE INFO |