Difference between revisions of "Subisomorph-containing subgroup"

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(Stronger properties)
(Relation with other properties)
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===Stronger properties===
 
===Stronger properties===
  
* [[Weaker than::Variety-containing subgroup]]
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{| class="sortable" border="1"
* [[Weaker than::Subhomomorph-containing subgroup]]: {{proofofstrictimplicationat|[[subhomomorph-containing implies subisomorph-containing]]|[[subisomorph-containing not implies subhomomorph-containing]]}}
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
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|-
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| [[Weaker than::variety-containing subgroup]] || contains every subgroup of the whole group in the variety generated by it || || || {{intermediate notions short|subisomorph-containing subgroup|variety-containing subgroup}}
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|-
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| [[Weaker than::subhomomorph-containing subgroup]] || contains every subgroup of the whole group isomorphic to a subquotient of it || || || {{intermediate notions short|subisomorph-containing subgroup|subhomomorph-containing subgroup}}
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|-
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| [[Weaker than::normal Sylow subgroup]] || normal subgroup of prime power order whose order and index are relatively prime || || || {{intermediate notions short|subisomorph-containing subgroup|normal Sylow subgroup}}
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|-
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| [[Weaker than::normal Hall subgroup]] || subgroup whose order and index are relatively prime || || || {{intermediate notions short|subisomorph-containing subgroup|normal Hall subgroup}}
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|}
  
 
===Weaker properties===
 
===Weaker properties===

Revision as of 13:54, 1 June 2020

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
If the ambient group is a finite group, this property is equivalent to the property: variety-containing subgroup
View other properties finitarily equivalent to variety-containing subgroup | View other variations of variety-containing subgroup |

Definition

A subgroup H of a group G is termed subisomorph-containing if whenever K is a subgroup of G and L is a subgroup of H such that K and L are isomorphic, then L is also a subgroup of H.

Relation with other properties

In groups with specific properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
variety-containing subgroup contains every subgroup of the whole group in the variety generated by it Subhomomorph-containing subgroup|FULL LIST, MORE INFO
subhomomorph-containing subgroup contains every subgroup of the whole group isomorphic to a subquotient of it |FULL LIST, MORE INFO
normal Sylow subgroup normal subgroup of prime power order whose order and index are relatively prime Variety-containing subgroup|FULL LIST, MORE INFO
normal Hall subgroup subgroup whose order and index are relatively prime Variety-containing subgroup|FULL LIST, MORE INFO

Weaker properties