Isomorph-containing subgroup

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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]

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Definition

QUICK PHRASES: contains all isomorphic subgroups, weakly closed in any ambient group

Equivalent definitions in tabular format

Below are many equivalent definitions of isomorph-containing subgroup.

No.	Shorthand	A subgroup of a group is isomorph-containing in it if...	A subgroup $H$ of a group $G$ is called an isomorph-containing subgroup of $G$ if ...
1	contains all isomorphic subgroups	it contains any subgroup of the whole group isomorphic to it.	for any subgroup $M$ of $G$ such that $H$ and $M$ are isomorphic groups, $M \le H$.
2	weakly closed in any ambient group	it is weakly closed in any ambient group of the whole group.	for any group $L$ containing $G$, $H$ is weakly closed in $G$ with respect to $L$.

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Equivalence of definitions

Further information: Isomorph-containing iff weakly closed in any ambient group

Examples

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Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
isomorph-free subgroup	no other isomorphic subgroup (for a finite subgroup, and more generally, for a co-Hopfian subgroup, the two properties are equivalent)	obvious	any example of a non-co-Hopfian group as a subgroup of itself -- such as the group of integers
homomorph-containing subgroup	contains any homomorphic image of itself in the group	obvious; isomorphs are also homomorphic images	cyclic maximal subgroup of dihedral group:D8 is isomorph-containing but not homomorph-containing	|FULL LIST, MORE INFO
Subhomomorph-containing subgroup	contains any homomorphic image in the whole group of any subgroup of it	(via homomorph-containing)	(via homomorph-containing)	Homomorph-containing subgroup, Right-transitively isomorph-containing subgroup, Subisomorph-containing subgroup|FULL LIST, MORE INFO
subisomorph-containing subgroup	contains any subgroup of the whole group isomorphic to any subgroup of itself	(obvious)	cyclic maximal subgroup of dihedral group:D8 is isomorph-containing but not subisomorph-containing	Right-transitively isomorph-containing subgroup|FULL LIST, MORE INFO
variety-containing subgroup	contains any subgroup of the whole group in the subvariety of the variety of groups generated by it	(obvious)	(via either homomorph-containing or subisomorph-containing)	Homomorph-containing subgroup, Right-transitively isomorph-containing subgroup, Subisomorph-containing subgroup|FULL LIST, MORE INFO
fully invariant direct factor	both a fully invariant subgroup and a direct factor	equivalence of definitions of fully invariant direct factor		Complemented homomorph-containing subgroup, Complemented isomorph-containing subgroup, Homomorph-containing subgroup|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
characteristic subgroup	invariant under any automorphism of the whole group	isomorph-containing implies characteristic	characteristic not implies isomorph-containing	Injective endomorphism-invariant subgroup, Intermediately characteristic subgroup, Intermediately injective endomorphism-invariant subgroup|FULL LIST, MORE INFO
injective endomorphism-invariant subgroup	invariant under any injective endomorphism of the whole group		can use same example as for characteristic not implies isomorph-containing, if finite	Intermediately injective endomorphism-invariant subgroup|FULL LIST, MORE INFO
intermediately injective endomorphism-invariant subgroup	injective endomorphism-invariant in every intermediate subgroup			|FULL LIST, MORE INFO
intermediately characteristic subgroup	characteristic in every intermediate subgroup			Intermediately injective endomorphism-invariant subgroup|FULL LIST, MORE INFO
normal subgroup	invariant under all inner automorphisms			Characteristic subgroup, Injective endomorphism-invariant subgroup, Intermediately characteristic subgroup|FULL LIST, MORE INFO
normal-isomorph-containing subgroup	contains any normal subgroup of the whole group isomorphic to it	(obvious)	any proper nontrivial subgroup of a finite simple non-abelian group	|FULL LIST, MORE INFO

Metaproperties

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

Here is a summary:

Metaproperty name	Satisfied?	Proof	Difficulty level (0-5)	Statement with symbols
transitive subgroup property	No	isomorph-containment is not transitive		It is possible to have groups $H \le K \le G$ such that $H$ is an isomorph-containing subgroup of $K$ and $K$ is an isomorph-containing subgroup of $G$, but $H$ is not an isomorph-containing subgroup of $G$.
trim subgroup property	Yes		0	In any group $G$, the trivial subgroup $\{ e \}$ and the whole group $G$ are both isomorph-containing subgroups of $G$.
intermediate subgroup condition	Yes		1	If $H \le K \le G$ are groups such that $H$ is isomorph-containing in $G$, then $H$ is also isomorph-containing in $K$.