# Finite supersolvable group

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

## Contents

## 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.
- 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.

## Examples

### Extreme examples

- The trivial group is a finite supersolvable group.

### Examples based on order

- Any group of prime power order is a finite nilpotent group and hence a finite supersolvable group.
- Any group whose order is square-free number is a metacyclic group and hence a finite supersolvable group.

### Other examples

- Any finite abelian group, and more generally, any finite nilpotent group, is a finite supersolvable group. Examples include any group of prime power order.
- Any metacyclic group is supersolvable, and therefore, any finite metacyclic group is a finite supersolvable group. Examples include all finite dihedral groups, dicyclic groups, as well as general affine group of degree one over a finite field when the field is a finite prime field (i.e., any group of the form for a prime number ).

### Non-examples

- The smallest example of a finite non-supersolvable group is alternating group:A4. In particular, it has no nontrivial cyclic normal subgroup, and therefore cannot be supersolvable. Therefore, the groups symmetric group:S4 and special linear group:SL(2,3) (which have as a subgroup and quotient respectively) are also non-supersolvable.
- In general, any finite non-solvable group is non-supersolvable. In particular, any group that contains a finite simple non-abelian group as a subgroup, quotient group, or subquotient is not supersolvable. Examples include the alternating group:A5 (the smallest finite simple non-abelian group), symmetric group:S5, and special linear group:SL(2,5)

## Metaproperties

Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|

subgroup-closed group property | Yes | If is a finite supersolvable group and is a subgroup of , then is also a finite supersolvable group. | |

quotient-closed group property | Yes | If is a finite supersolvable group and is a normal subgroup of , then the quotient group is also a finite supersolvable group. | |

finite direct product-closed group property | Yes | If are all finite supersolvable groups, the external direct product is also a finite supersolvable group. | |

lattice-determined group property | Yes | follows from characterization as finite J-group | If 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 | Finite nilpotent group|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 |