# 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.
4. It is finite and a J-group, i.e., its lattice of subgroups satisfies the Jordan-Dedekind chain condition: all chains between two subgroups (with one contained in the other) have equal length.

## Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
subgroup-closed group property Yes If $G$ is a finite supersolvable group and $H$ is a subgroup of $G$, then $H$ is also a finite supersolvable group.
quotient-closed group property Yes If $G$ is a finite supersolvable group and $H$ is a normal subgroup of $G$, then the quotient group $G/H$ is also a finite supersolvable group.
finite direct product-closed group property Yes If $G_1, G_2, \dots, G_n$ are all finite supersolvable groups, the external direct product $G_1 \times G_2 \times \dots \times G_n$ is also a finite supersolvable group.
lattice-determined group property Yes follows from characterization as finite J-group If $G_1, G_2$ have isomorphic lattices of subgroups, then either both are finite supersolvable, or neither is.

## 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 Group having subgroups of all orders dividing the group order|FULL LIST, MORE INFO