Normal subgroup of nilpotent group: Difference between revisions
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The term '''normal subgroup of nilpotent group''' is used for a [[subgroup]] of a [[group]] where the whole group is a [[nilpotent group]] and the subgroup is a [[normal subgroup]]. | The term '''normal subgroup of nilpotent group''' is used for a [[subgroup]] of a [[group]] where the whole group is a [[nilpotent group]] and the subgroup is a [[normal subgroup]]. | ||
==Relation with other properties== | |||
===Stronger properties=== | |||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Weaker than::subgroup of abelian group]] || || || || | |||
|- | |||
| [[Weaker than::normal subgroup of group of prime power order]] || || || || | |||
|- | |||
| [[Weaker than::characteristic subgroup of nilpotent group]] || || || || | |||
|} | |||
===Weaker properties=== | |||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Stronger than::normal subgroup]] || || || || | |||
|- | |||
| [[Stronger than::subgroup of nilpotent group]] || || || || | |||
|- | |||
| [[Stronger than::unipotent automorphism-invariant subgroup]] || || || || | |||
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Revision as of 23:52, 23 June 2012
This article describes a property that arises as the conjunction of a subgroup property: normal subgroup with a group property imposed on the ambient group: nilpotent group
View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup
Definition
The term normal subgroup of nilpotent group is used for a subgroup of a group where the whole group is a nilpotent group and the subgroup is a normal subgroup.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| subgroup of abelian group | ||||
| normal subgroup of group of prime power order | ||||
| characteristic subgroup of nilpotent group |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| normal subgroup | ||||
| subgroup of nilpotent group | ||||
| unipotent automorphism-invariant subgroup |