Isomorph-free subgroup

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Definition

QUICK PHRASES: no other isomorphic subgroups, no isomorphic copies, only subgroup of its isomorphism type

Symbol-free definition

A subgroup of a group is said to be isomorph-free if it satisfies the following equivalent conditions:

1. There is no other subgroup of the group isomorphic to it as an abstract group.
2. It is an isomorph-containing subgroup that is also a co-Hopfian group (in other words, it contains every subgroup of the whole group isomorphic to it, but no proper subgroup of it is isomorphic to it).

Definition with symbols

A subgroup $H$ of a group $G$ is said to be isomorph-free if it satisfies the following equivalent conditions:

1. Whenever $K\le G$ such that $H\cong K$, then $H=K$ (i.e. $H$ and $K$ are the same subgroup).
2. $H$ is a co-Hopfian group, and whenever $K\le G$ such that $H\cong K$, then $K\le H$.

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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VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

This is a variation of characteristicity|Find other variations of characteristicity | Read a survey article on varying characteristicity

This article describes a property that arises as the conjunction of a subgroup property: isomorph-containing subgroup with a group property (itself viewed as a subgroup property): co-Hopfian group
View a complete list of such conjunctions

Examples

Extreme examples

• The trivial subgroup is isomorph-free.
• Any co-Hopfian group (and in particular, any finite group) is isomorph-free as a subgroup of itself.

Examples in small finite groups

Below are some examples of a proper nontrivial subgroup that satisfy the property isomorph-free subgroup.

Below are some examples of a proper nontrivial subgroup that does not satisfy the property isomorph-free subgroup.

	Group part	Subgroup part	Quotient part
Center of dihedral group:D8	Dihedral group:D8	Cyclic group:Z2	Klein four-group

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
normal Sylow subgroup	Sylow subgroup that is also normal	Follows from Sylow implies order-conjugate		|FULL LIST, MORE INFO
normal Hall subgroup	Hall subgroup (i.e., order and index are relatively prime) that is also normal			|FULL LIST, MORE INFO
order-unique subgroup	unique subgroup of its order		isomorph-free not implies order-unique (see also list of examples)	|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
isomorph-containing subgroup	contains every isomorphic subgroup (note that this property is equivalent when the subgroup is finite or more generally a co-Hopfian group)	obvious	any example of a non-co-Hopfian group as a subgroup of itself -- such as the group of integers
characteristic subgroup	invariant under all automorphisms	(via isomorph-containing)	characteristic not implies isomorph-free in finite	Isomorph-normal characteristic 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 all intermediate subgroups			|FULL LIST, MORE INFO
intermediately characteristic subgroup	characteristic in every intermediate subgroup		(via isomorph-containing) (see also list of examples)	|FULL LIST, MORE INFO
normal subgroup	invariant under all inner automorphisms	(via characteristic)	(via characteristic) (see also list of examples)	Characteristic subgroup, Hall-relatively weakly closed subgroup, Isomorph-normal characteristic subgroup, Isomorph-normal subgroup|FULL LIST, MORE INFO
isomorph-conjugate subgroup	all isomorphic subgroups are conjugate to it	obvious	(see also list of examples)	|FULL LIST, MORE INFO
intermediately isomorph-conjugate subgroup	isomorph-conjugate in every intermediate subgroup			|FULL LIST, MORE INFO
automorph-conjugate subgroup	all automorphic subgroups are conjugate to it	(via isomorph-conjugate)	(via isomorph-conjugate)	Characteristic subgroup|FULL LIST, MORE INFO
intermediately automorph-conjugate subgroup	automorph-conjugate in every intermediate subgroup			|FULL LIST, MORE INFO
normal-isomorph-free subgroup	normal and no other isomorphic normal subgroup			|FULL LIST, MORE INFO
characteristic-isomorph-free subgroup	characteristic and no other isomorphic characteristic subgroup			|FULL LIST, MORE INFO
series-isomorph-free subgroup	normal and no isomorphic normal subgroup with isomorphic quotient			|FULL LIST, MORE INFO
isomorph-normal subgroup	every isomorphic subgroup is normal		(see also list of examples)	Isomorph-characteristic subgroup, Isomorph-normal characteristic subgroup|FULL LIST, MORE INFO
isomorph-characteristic subgroup	every isomorphic subgroup is characteristic		(see also list of examples)	|FULL LIST, MORE INFO
isomorph-normal characteristic subgroup	characteristic and every isomorphic subgroup is normal			|FULL LIST, MORE INFO

Metaproperties

Transitivity

NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole group
ABOUT THIS PROPERTY: View variations of this property that are transitive|View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of subgroup properties that are not transitive|View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity

An isomorph-free subgroup of an isomorph-free subgroup need not be isomorph-free. Further information: Isomorph-freeness is not transitive

Quotient-transitivity

This subgroup property is quotient-transitive: the corresponding quotient property is transitive.
View a complete list of quotient-transitive subgroup properties

If $H$ is an isomorph-free subgroup of $G$ and $K/H$ is an isomorph-free subgroup of $G/H$, then $K$ is an isomorph-free subgroup of $G$.

For full proof, refer: Isomorph-freeness is quotient-transitive

Intersection-closedness

This subgroup property is not intersection-closed, viz., it is not true that an intersection of subgroups with this property must have this property.
Read an article on methods to prove that a subgroup property is not intersection-closed

If $H,K$ are isomorph-free subgroups of $G$, the intersection $H\cap K$ need not be isomorph-free. For full proof, refer: Isomorph-freeness is not intersection-closed

Join-closedness

YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are join-closed | View variations of this property that are not join-closed
ABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness

If $H_i,i\in I$ is a collection of isomorph-free subgroups of $G$, the join of the $H_i$s is also isomorph-free.

For full proof, refer: Isomorph-freeness is strongly join-closed

Upper join-closedness

NO: This subgroup property is not upper join-closed: if a subgroup has the property in intermediate subgroups it need not have the property in their join.

If $H\le G$ and $K,L$ are intermediate subgroups such that $H$ is isomorph-free in both $K$ and $L$, $H$ need not be isomorph-free in $⟨K,L⟩$. For full proof, refer: Isomorph-freeness is not upper join-closed

Trimness

The property of being isomorph-free is trivially true, viz., it is satisfied by the trivial subgroup. However, a group need not be isomorph-free in itself, because it may be isomorphic to a proper subgroup of itself (the condition of being isomorph-free as a subgroup of itself, is precisely the condition of being a co-Hopfian group).

Effect of property operators

The subordination operator

Applying the subordination operator to this property gives: sub-isomorph-free subgroup

A subgroup $H$ of a group $G$ is termed sub-isomorph-free if there is a series of subgroups $H=H_0\le H_1\le \dots \le H_n=G$, with each $H_{i-1}$ an isomorph-free subgroup of $H_i$.

Testing

GAP code

One can write code to test this subgroup property in GAP (Groups, Algorithms and Programming), though there is no direct command for it.
View the GAP code for testing this subgroup property at: IsIsomorphFreeSubgroup
View other GAP-codable subgroup properties | View subgroup properties with in-built commands

GAP-codable subgroup property

While there is no in-built command for testing whether a subgroup is isomorph-free, a short piece of GAP code can do the test. The code can be found at GAP:IsIsomorphFreeSubgroup, and the command is invoked as follows:

IsIsomorphFreeSubgroup(group,subgroup);