Group generated by abelian normal subgroups: Difference between revisions
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===Symbol-free definition=== | ===Symbol-free definition=== | ||
A [[group]] is said to be '''generated by | A [[group]] is said to be '''generated by abelian normal subgroups''' if there exists a collection of [[abelian normal subgroup]]s which together generate the group. | ||
==Examples== | ==Examples== | ||
The [[dihedral group: | The [[dihedral group:D8|dihedral group of size eight]] and the [[quaternion group]] are examples of non-Abelian groups generated by Abelian normal subgroups. While the former is generated by a [[cyclic normal subgroup]] of order 4 and a [[Klein four-group]], the latter is generated by two cyclic normal subgroups. | ||
==Relation with other properties== | ==Relation with other properties== | ||
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===Stronger properties=== | ===Stronger properties=== | ||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Weaker than::abelian group]] || nilpotency class at most one || (obvious) || || {{intermediate notions short|group generated by abelian normal subgroups|abelian group}} | |||
|- | |||
| [[Weaker than::group of nilpotency class two]] || nilpotency class at most two || [[class two implies generated by abelian normal subgroups]] || [[finite group generated by abelian normal subgroups may have arbitrarily large nilpotency class]] || {{intermediate notions short|group generated by abelian normal subgroups|group of nilpotency class two}} | |||
|- | |||
| [[Weaker than::Levi group]] || every element is in an [[abelian normal subgroup]], equivalently, a 2-[[Engel group]] || [[Levi implies generated by abelian normal subgroups]] || || {{intermediate notions short|group generated by abelian normal subgroups|Levi group}} | |||
|} | |||
===Weaker properties=== | ===Weaker properties=== | ||
* [[ | * [[nilpotent group]] (for [[finite group]]s): {{proofofstrictimplicationat|[[Finite and generated by abelian normal subgroups implies nilpotent]]|[[Nilpotent not implies generated by abelian normal subgroups]]}} | ||
==Facts== | |||
* [[Finite group generated by abelian normal subgroups may have arbitrarily large nilpotency class]] | |||
Latest revision as of 15:11, 2 August 2011
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
Definition
Symbol-free definition
A group is said to be generated by abelian normal subgroups if there exists a collection of abelian normal subgroups which together generate the group.
Examples
The dihedral group of size eight and the quaternion group are examples of non-Abelian groups generated by Abelian normal subgroups. While the former is generated by a cyclic normal subgroup of order 4 and a Klein four-group, the latter is generated by two cyclic normal subgroups.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| abelian group | nilpotency class at most one | (obvious) | 2-Engel group|FULL LIST, MORE INFO | |
| group of nilpotency class two | nilpotency class at most two | class two implies generated by abelian normal subgroups | finite group generated by abelian normal subgroups may have arbitrarily large nilpotency class | 2-Engel group|FULL LIST, MORE INFO |
| Levi group | every element is in an abelian normal subgroup, equivalently, a 2-Engel group | Levi implies generated by abelian normal subgroups | |FULL LIST, MORE INFO |
Weaker properties
- nilpotent group (for finite groups): For proof of the implication, refer Finite and generated by abelian normal subgroups implies nilpotent and for proof of its strictness (i.e. the reverse implication being false) refer Nilpotent not implies generated by abelian normal subgroups.