Difference between revisions of "Finite solvable group"

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(Examples)
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* The [[trivial group]] is a finite solvable group.
 
* 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
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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.
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The smallest order examples of finite non-solvable groups are below:
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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]]
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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 not simple.
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* [[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==

Revision as of 16:00, 7 July 2011

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.

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

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