# Difference between revisions of "Finite solvable group"

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* Any product of two prime powers, i.e., any number of the form <math>p^aq^b</math>, with <math>p,q</math> primes, is solvability-forcing. See [[order has only two prime factors implies solvable]] (this result is also termed Burnside's <math>p^aq^b</math>-theorem). | * Any product of two prime powers, i.e., any number of the form <math>p^aq^b</math>, with <math>p,q</math> primes, is solvability-forcing. See [[order has only two prime factors implies solvable]] (this result is also termed Burnside's <math>p^aq^b</math>-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 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 | + | * Any [[square-free number]] i.e., any number that is a product of pairwise distinct primes. See [[square-free implies solvability-forcing]]. |

===Non-examples=== | ===Non-examples=== | ||

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| [[polycyclic group]] || has a [[subnormal series]] (of finite length) with cyclic quotient groups || stronger than solvability || {{intermediate notions short|solvable group|polycyclic group}} | | [[polycyclic group]] || has a [[subnormal series]] (of finite length) with cyclic quotient groups || stronger than solvability || {{intermediate notions short|solvable group|polycyclic group}} | ||

|- | |- | ||

− | | [[locally solvable group]] || every [[finitely generated group|finitely generated]] subgroup is solvable || weaker than solvability | {{intermediate notions short|locally solvable group|solvable group}} | + | | [[locally solvable group]] || every [[finitely generated group|finitely generated]] subgroup is solvable || weaker than solvability || {{intermediate notions short|locally solvable group|solvable group}} |

|- | |- | ||

− | | [[hypoabelian group]] || the transfinite [[derived series]] reaches the identity element | + | | [[hypoabelian group]] || the transfinite [[derived series]] reaches the identity element || weaker than solvability || {{intermediate notions short|hypoabelian group|solvable group}} |

|- | |- | ||

| [[residually solvable group]] || every non-identity element is outside a [[normal subgroup]] for which the [[quotient group]] is solvable || weaker than solvability || {{intermediate notions short|residually solvable group|solvable group}} | | [[residually solvable group]] || every non-identity element is outside a [[normal subgroup]] for which the [[quotient group]] is solvable || weaker than solvability || {{intermediate notions short|residually solvable group|solvable group}} | ||

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|- | |- | ||

| [[Weaker than::finite supersolvable group]] || finite and a [[supersolvable group]] || follows from [[supersolvable implies solvable]] || see [[solvable not implies supersolvable]] || {{intermediate notions short|finite solvable group|finite supersolvable group}} | | [[Weaker than::finite supersolvable group]] || finite and a [[supersolvable group]] || follows from [[supersolvable implies solvable]] || see [[solvable not implies supersolvable]] || {{intermediate notions short|finite solvable group|finite supersolvable group}} | ||

+ | |- | ||

+ | | [[Weaker than::group having subgroups of all orders dividing the group order]] || finite and has subgroups of all orders dividing the group order || [[subgroups of all orders dividing the group order implies solvable]] || || {{intermediate notions short|finite solvable group|group having subgroups of all orders dividing the group order}} | ||

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## Latest revision as of 02:45, 26 December 2015

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 |