Difference between revisions of "Abelian pronormal subgroup"

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(Relation with other properties)
(Weaker properties)
 
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| [[Stronger than::central subgroup of pronormal subgroup]] || [[central subgroup]] of a [[pronormal subgroup]] || || || {{intermediate notions short|central subgroup of pronormal subgroup|abelian pronormal subgroup}}
 
| [[Stronger than::central subgroup of pronormal subgroup]] || [[central subgroup]] of a [[pronormal subgroup]] || || || {{intermediate notions short|central subgroup of pronormal subgroup|abelian pronormal subgroup}}
 
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| [[Stronger than::SCDIN-subgroup]] || || [[abelian and pronormal implies SCDIN]] (also, via center of pronormal subgroup) || || {{intermediate notions short|SCDIN-subgroup|abelian pronormal subgroup}}
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| [[Stronger than::SCDIN-subgroup]] || [[subset-conjugacy-determined subgroup]] in its [[normalizer]]  || [[abelian and pronormal implies SCDIN]] (also, via center of pronormal subgroup) || || {{intermediate notions short|SCDIN-subgroup|abelian pronormal subgroup}}
 
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| [[Stronger than::CDIN-subgroup]] || || (via SCDIN) || || {{intermediate notions short|CDIN-subgroup|abelian pronormal subgroup}}
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| [[Stronger than::CDIN-subgroup]] || [[conjugacy-determined subgroup]] in its [[normalizer]] || (via SCDIN) || || {{intermediate notions short|CDIN-subgroup|abelian pronormal subgroup}}
 
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| [[Stronger than::pronormal subgroup]] || || (by definition) || || {{intermediate notions short|pronormal subgroup|abelian pronormal subgroup}}
 
| [[Stronger than::pronormal subgroup]] || || (by definition) || || {{intermediate notions short|pronormal subgroup|abelian pronormal subgroup}}
 
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Latest revision as of 23:15, 11 July 2010

This article describes a property that arises as the conjunction of a subgroup property: pronormal subgroup with a group property (itself viewed as a subgroup property): abelian group
View a complete list of such conjunctions

Definition

A subgroup H of a group G is termed an abelian pronormal subgroup of G if it satisfies the following two conditions:

  1. H is an abelian group.
  2. H is a pronormal subgroup of G: If K is a conjugate of H in G, then H, K are also conjugate in \langle H, K \rangle.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
central subgroup contained in the center |FULL LIST, MORE INFO
abelian normal subgroup both abelian as a group and a normal subgroup |FULL LIST, MORE INFO
abelian Sylow subgroup both abelian as a group and a Sylow subgroup |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
center of pronormal subgroup center of a pronormal subgroup any abelian pronormal subgroup is its own center |FULL LIST, MORE INFO
central subgroup of pronormal subgroup central subgroup of a pronormal subgroup Center of pronormal subgroup|FULL LIST, MORE INFO
SCDIN-subgroup subset-conjugacy-determined subgroup in its normalizer abelian and pronormal implies SCDIN (also, via center of pronormal subgroup) Center of pronormal subgroup|FULL LIST, MORE INFO
CDIN-subgroup conjugacy-determined subgroup in its normalizer (via SCDIN) SCDIN-subgroup|FULL LIST, MORE INFO
pronormal subgroup (by definition) |FULL LIST, MORE INFO