Difference between revisions of "Finite solvable group"

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# It has [[Hall subgroup]]s of all possible orders
 
# It has [[Hall subgroup]]s of all possible orders
 
# All its composition factors (i.e., the quotient groups for any [[defining ingredient::composition series]] for the group) are cyclic groups of prime order.
 
# All its composition factors (i.e., the quotient groups for any [[defining ingredient::composition series]] for the group) are cyclic groups of prime order.
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==Examples==
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===Extreme examples===
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* The [[trivial group]] is a finite solvable group.
  
 
==Relation with other properties==
 
==Relation with other properties==
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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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|}

Revision as of 21:29, 26 May 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.

Examples

Extreme examples

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