# Difference between revisions of "Isomorph-containing subgroup"

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## Contents

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]

## Definition

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

A subgroup $H$ of a group $G$ is termed an isomorph-containing subgroup if it satisfies the following equivalent conditions:

1. Whenever $K \le G$ is a subgroup of $G$ isomorphic to $H$, $K \le H$.
2. If $G$ is a subgroup of $L$, $H$ is weakly closed in $G$ with respect to $L$.

### Equivalence of definitions

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

## Examples

VIEW: subgroups of groups satisfying this property | subgroups of groups dissatisfying this property
VIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions

## 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 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|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 $K$