# Finite solvable 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

## Contents

## Definition

A finite group is termed a **finite solvable group** if it satisfies the following equivalent conditions:

- It is a solvable group
- It is a polycyclic group
- It has Sylow complements for all prime divisors of the order of the group
- It has Hall subgroups of all possible orders
- All its composition factors (i.e., the quotient groups for any composition series for the group) are cyclic groups of prime order. Equivalently, all its composition factors are abelian.
- All its chief factors (i.e., the successive quotient groups for any chief series for the group) are elementary abelian groups.

### Equivalence of definitions

`Further information: equivalence of definitions of finite solvable group`

## Examples

### Extreme examples

- The trivial group is a finite solvable group.

### Examples based on order

We call a natural number a solvability-forcing number if every group of order is solvable. It turns out that:

- Any prime power is solvability-forcing, because prime power order implies nilpotent and nilpotent implies solvable.
- Any product of two prime powers, i.e., any number of the form , with primes, is solvability-forcing. See order has only two prime factors implies solvable (this result is also termed Burnside's -theorem).
- Any odd number is solvability-forcing. See odd-order implies solvable. This result, also called the
**odd-order theorem**or the Feit-Thompson theorem, is highly nontrivial. - Any square-free number i.e., any number that is a product of pairwise distinct primes. See square-free implies solvability-forcing.

### Non-examples

Any finite simple non-abelian group is a finite group that is not solvable. See classification of finite simple groups for a list of finite simple non-abelian groups.

Further, any group that contains a finite simple non-abelian group as a subgroup, has a finite simple non-abelian group as a quotient group, or admits a finite simple non-abelian group as a subquotient must be non-solvable.

The smallest order examples of finite non-solvable groups are below:

- alternating group:A5: This is the smallest order simple non-abelian group. It has order 60. See A5 is the simple non-abelian group of smallest order
- symmetric group:S5, special linear group:SL(2,5), direct product of A5 and Z2: All of these are groups of order 120 which have alternating group:A5 as one of their composition factors, and are hence neither simple nor solvable.
- projective special linear group:PSL(3,2): This is a simple non-abelian group of order 168.

## Relation with other properties

### Properties whose conjunction with finiteness gives this property

Below is a list of group properties such that a finite group has the property if and only if it is a finite solvable group.

Property | Meaning | Relation with solvability in general | Intermediate properties between it and solvability in general |
---|---|---|---|

solvable group | derived series reaches identity in finitely many steps | same | -- |

polycyclic group | has a subnormal series (of finite length) with cyclic quotient groups | stronger than solvability | Finitely generated solvable group, Finitely presented solvable group|FULL LIST, MORE INFO |

locally solvable group | every finitely generated subgroup is solvable | weaker than solvability | |FULL LIST, MORE INFO |

hypoabelian group | the transfinite derived series reaches the identity element | weaker than solvability | Residually solvable group|FULL LIST, MORE INFO |

residually solvable group | every non-identity element is outside a normal subgroup for which the quotient group is solvable | weaker than solvability | |FULL LIST, MORE INFO |

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

finite abelian group | finite and an abelian group | follows from abelian implies solvable | see solvable not implies abelian | Finite nilpotent group, Finite supersolvable group, Group having subgroups of all orders dividing the group order|FULL LIST, MORE INFO |

finite nilpotent group | finite and a nilpotent group | follows from nilpotent implies solvable | see solvable not implies nilpotent | Finite supersolvable group, Group having a Sylow tower, Group having subgroups of all orders dividing the group order|FULL LIST, MORE INFO |

finite supersolvable group | finite and a supersolvable group | follows from supersolvable implies solvable | see solvable not implies supersolvable | Group having subgroups of all orders dividing the group order|FULL LIST, MORE INFO |