Subisomorph-containing subgroup: Difference between revisions

Latest revision as of 14:06, 1 June 2020

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If the ambient group is a finite group, this property is equivalent to the property: variety-containing subgroup
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Definition

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

Relation with other properties

In groups with specific properties

Finite group and periodic group: In a finite group and more generally in a periodic group, the notion of subisomorph-containing subgroup coincides with the notions of subhomomorph-containing subgroup and variety-containing subgroup. Further information: Equivalence of definitions of variety-containing subgroup of finite group, Equivalence of definitions of variety-containing subgroup of periodic group
Group of prime power order: For groups of prime power order, subisomorph-containing subgroups must be omega subgroups of group of prime power order, though the converse, while true for regular p-groups, is not always true. For full proof, refer: Variety-containing implies omega subgroup in group of prime power order, omega subgroups are variety-containing in regular p-group, omega subgroups not are variety-containing

Stronger properties

Property	Meaning	Intermediate notions
variety-containing subgroup	contains every subgroup of the whole group in the variety generated by it	\|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	\|FULL LIST, MORE INFO
normal Hall subgroup	subgroup whose order and index are relatively prime	\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Intermediate notions
isomorph-containing subgroup	contains every isomorphic subgroup	\|FULL LIST, MORE INFO
injective endomorphism-invariant subgroup	invariant under all injective endomorphisms	\|FULL LIST, MORE INFO
intermediately injective endomorphism-invariant subgroup	injective endomorphism-invariant in every intermediate subgroup	\|FULL LIST, MORE INFO
characteristic subgroup	invariant under every automorphism	\|FULL LIST, MORE INFO
intermediately characteristic subgroup	characteristic in every intermediate subgroup	\|FULL LIST, MORE INFO
transfer-closed characteristic subgroup	intersection with any subgroup is characteristic in that subgroup	\|FULL LIST, MORE INFO

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 {{wikilocal}}
 {{subgroup property}}
+{{finitarily equivalent to|variety-containing subgroup}}
 ==Definition==
-A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''subisomorph-containing''' if whenever <math>K</math> is a subgroup of <math>G</math> and <math>L</math> is a subgroup of <math>H</math> such that <math>K</math> and <math>L</math> are [[isomorphic groups|isomorphic]], then <math>L</math> is also a subgroup of <math>H</math>.
+A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''subisomorph-containing''' if whenever <math>K</math> is a subgroup of <math>H</math> and <math>L</math> is a subgroup of <math>G</math> such that <math>K</math> and <math>L</math> are [[isomorphic groups|isomorphic]], then <math>L</math> is also a subgroup of <math>H</math>.
 ==Relation with other properties==
+===In groups with specific properties===
+* [[Finite group]] and [[periodic group]]: In a finite group and more generally in a [[periodic group]], the notion of subisomorph-containing subgroup coincides with the notions of [[subhomomorph-containing subgroup]] and [[variety-containing subgroup]]. {{further|[[Equivalence of definitions of variety-containing subgroup of finite group]], [[Equivalence of definitions of variety-containing subgroup of periodic group]]}}
+* [[Group of prime power order]]: For groups of prime power order, subisomorph-containing subgroups must be [[omega subgroups of group of prime power order]], though the converse, while true for [[regular p-group]]s, is not always true. {{proofat|[[Variety-containing implies omega subgroup in group of prime power order]], [[omega subgroups are variety-containing in regular p-group]], [[omega subgroups not are variety-containing]]}}
 ===Stronger properties===
-* [[Weaker than::Variety-containing subgroup]]
+{| class="sortable" border="1"
-* [[Weaker than::Subhomomorph-containing subgroup]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[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}}
+|-
+| [[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}}
+|-
+| [[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}}
+|-
+| [[Weaker than::normal Hall subgroup]] || subgroup whose order and index are relatively prime || || || {{intermediate notions short|subisomorph-containing subgroup|normal Hall subgroup}}
+|}
 ===Weaker properties===
-* [[Stronger than::Isomorph-containing subgroup]]
+{| class="sortable" border="1"
-* [[Stronger than::Injective endomorphism-invariant subgroup]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Stronger than::Intermediately injective endomorphism-invariant subgroup]]
+|-
-* [[Stronger than::Intermediately characteristic subgroup]]
+| [[Stronger than::isomorph-containing subgroup]] || contains every isomorphic subgroup || || || {{intermediate notions short|isomorph-containing subgroup|subisomorph-containing subgroup}}
-* [[Stronger than::Characteristic subgroup]]
+|-
+| [[Stronger than::injective endomorphism-invariant subgroup]] || invariant under all injective endomorphisms || || || {{intermediate notions short|injective endomorphism-invariant subgroup|subisomorph-containing subgroup}}
+|-
+| [[Stronger than::intermediately injective endomorphism-invariant subgroup]] || injective endomorphism-invariant in every intermediate subgroup || || || {{intermediate notions short|intermediately injective endomorphism-invariant subgroup|subisomorph-containing subgroup}}
+|-
+| [[Stronger than::characteristic subgroup]] || invariant under every automorphism || || || {{intermediate notions short|characteristic subgroup|subisomorph-containing subgroup}}
+|-
+| [[Stronger than::intermediately characteristic subgroup]] || characteristic in every intermediate subgroup || || || {{intermediate notions short|intermediately characteristic subgroup|subisomorph-containing subgroup}}
+|-
+| [[Stronger than::transfer-closed characteristic subgroup]] || intersection with any subgroup is characteristic in that subgroup || || || {{intermediate notions short|transfer-closed characteristic subgroup|subisomorph-containing subgroup}}
+|}