Difference between revisions of "Finite solvable group"

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{{finite group property}}
 
{{finite group property}}
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[[importance rank::2| ]]
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==Definition==
  
==Definition==
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A [[defining ingredient::finite group]] is termed a '''finite solvable group''' if it satisfies the following equivalent conditions:
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# It is a [[defining ingredient::solvable group]]
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# It is a [[defining ingredient::polycyclic group]]
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# It has [[Sylow complement]]s for all prime divisors of the order of the group
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# It has [[Hall subgroup]]s of all possible orders
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# 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.
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# 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.
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===Equivalence of definitions===
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{{further|[[equivalence of definitions of finite solvable group]]}}
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==Examples==
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===Extreme examples===
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* The [[trivial group]] is a finite solvable group.
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===Examples based on order===
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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:
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* Any [[prime power]] is solvability-forcing, because [[prime power order implies nilpotent]] and [[nilpotent implies solvable]].
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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).
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* 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.
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* Any [[square-free number]] i.e., any number that is a product of pairwise distinct primes. See [[square-free implies solvability-forcing]].
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===Non-examples===
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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.
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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.
  
A [[finite group]] is termed a '''finite solvable group''' if it satisfies the following equivalent conditions:
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The smallest order examples of finite non-solvable groups are below:
  
* It is a [[solvable group]]
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* [[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]]
* It is a [[polycyclic group]]
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* [[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.
* It has [[Sylow complement]]s for all prime divisors of the order of the group
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* [[projective special linear group:PSL(3,2)]]: This is a simple non-abelian group of order 168.
* It has [[Hall subgroup]]s of all possible orders
 
  
 
==Relation with other properties==
 
==Relation with other properties==
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===Properties whose conjunction with finiteness gives this property===
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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]].
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{| class="sortable" border="1"
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! Property !! Meaning !! Relation with solvability in general !! Intermediate properties between it and solvability in general
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|-
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| [[solvable group]] || [[derived series]] reaches identity in finitely many steps || same || --
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|-
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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}}
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|-
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| [[locally solvable group]] || every [[finitely generated group|finitely generated]] subgroup is solvable || weaker than solvability || {{intermediate notions short|locally solvable group|solvable group}}
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|-
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| [[hypoabelian group]] || the transfinite [[derived series]] reaches the identity element || weaker than solvability || {{intermediate notions short|hypoabelian group|solvable group}}
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|-
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| [[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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|}
  
 
===Stronger properties===
 
===Stronger properties===
  
* [[Finite Abelian group]]
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{| class="sortable" border="1"
* [[Finite nilpotent group]]
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
* [[Finite supersolvable group]]
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|-
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| [[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}}
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|-
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| [[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}}
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|-
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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}}
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|-
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| [[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

Definition

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

  1. It is a solvable group
  2. It is a polycyclic group
  3. It has Sylow complements for all prime divisors of the order of the group
  4. It has Hall subgroups of all possible orders
  5. 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.
  6. 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

Examples based on order

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

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:

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
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 |FULL LIST, MORE INFO