# Difference between revisions of "Finite solvable group"

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{{finite group property}} | {{finite group property}} | ||

+ | [[importance rank::2| ]] | ||

+ | ==Definition== | ||

− | == | + | A [[defining ingredient::finite group]] is termed a '''finite solvable group''' if it satisfies the following equivalent conditions: |

+ | |||

+ | # It is a [[defining ingredient::solvable group]] | ||

+ | # It is a [[defining ingredient::polycyclic group]] | ||

+ | # It has [[Sylow complement]]s for all prime divisors of the order of the group | ||

+ | # It has [[Hall subgroup]]s of all possible orders | ||

+ | # All its [[composition factor]]s (i.e., the quotient groups for any [[defining ingredient::composition series]] for the group) are cyclic groups of prime order. Equivalently, all its composition factors are abelian. | ||

+ | # All its [[chief factor]]s (i.e., the successive quotient groups for any [[defining ingredient::chief series]] for the group) are [[elementary abelian group]]s. | ||

+ | |||

+ | ===Equivalence of definitions=== | ||

+ | |||

+ | {{further|[[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 <math>n</math> a [[solvability-forcing number]] if every group of order <math>n</math> 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 <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 [[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: | |

− | * It | + | * [[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 factor]]s, 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== | ==Relation with other properties== | ||

+ | |||

+ | ===Properties whose conjunction with finiteness gives this property=== | ||

+ | |||

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

+ | |||

+ | {| class="sortable" border="1" | ||

+ | ! 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 || {{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}} | ||

+ | |- | ||

+ | | [[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}} | ||

+ | |} | ||

===Stronger properties=== | ===Stronger properties=== | ||

− | + | {| class="sortable" border="1" | |

− | + | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |

− | + | |- | |

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

+ | |- | ||

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

+ | |} |

## 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 |