# Isomorph-free subgroup

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## Definition

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

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]
This is a variation of characteristic subgroup|Find other variations of characteristic subgroup | Read a survey article on varying characteristic subgroup
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.

Group partSubgroup partQuotient part
A3 in S3Symmetric group:S3Cyclic group:Z3Cyclic group:Z2
A4 in S4Symmetric group:S4Alternating group:A4Cyclic group:Z2
Center of nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Klein four-groupKlein four-group
Center of quaternion groupQuaternion groupCyclic group:Z2Klein four-group
Cyclic maximal subgroup of dihedral group:D16Dihedral group:D16Cyclic group:Z8Cyclic group:Z2
Cyclic maximal subgroup of dihedral group:D8Dihedral group:D8Cyclic group:Z4Cyclic group:Z2
Cyclic maximal subgroup of semidihedral group:SD16Semidihedral group:SD16Cyclic group:Z8Cyclic group:Z2
D8 in SD16Semidihedral group:SD16Dihedral group:D8Cyclic group:Z2
Derived subgroup of dihedral group:D16Dihedral group:D16Cyclic group:Z4Klein four-group
Direct product of Z4 and Z2 in M16M16Direct product of Z4 and Z2Cyclic group:Z2
First omega subgroup of direct product of Z4 and Z2Direct product of Z4 and Z2Klein four-groupCyclic group:Z2
Klein four-subgroup of M16M16Klein four-groupCyclic group:Z4
Klein four-subgroup of alternating group:A4Alternating group:A4Klein four-groupCyclic group:Z3
Q8 in SD16Semidihedral group:SD16Quaternion groupCyclic group:Z2
Q8 in central product of D8 and Z4Central product of D8 and Z4Quaternion groupCyclic group:Z2
SL(2,3) in GL(2,3)General linear group:GL(2,3)Special linear group:SL(2,3)Cyclic group:Z2

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

Group partSubgroup partQuotient part
2-Sylow subgroup of general linear group:GL(2,3)General linear group:GL(2,3)Semidihedral group:SD16
A3 in A4Alternating group:A4Cyclic group:Z3
A3 in A5Alternating group:A5Cyclic group:Z3
A3 in S4Symmetric group:S4Cyclic group:Z3
A4 in A5Alternating group:A5Alternating group:A4
Center of M16M16Cyclic group:Z4Klein four-group
Center of dihedral group:D16Dihedral group:D16Cyclic group:Z2Dihedral group:D8
Center of dihedral group:D8Dihedral group:D8Cyclic group:Z2Klein four-group
Center of semidihedral group:SD16Semidihedral group:SD16Cyclic group:Z2Dihedral group:D8
Center of unitriangular matrix group:UT(3,p)Unitriangular matrix group:UT(3,p)Group of prime orderElementary abelian group of prime-square order
Cyclic maximal subgroups of quaternion groupQuaternion groupCyclic group:Z4Cyclic group:Z2
D8 in A6Alternating group:A6Dihedral group:D8
D8 in D16Dihedral group:D16Dihedral group:D8Cyclic group:Z2
D8 in S4Symmetric group:S4Dihedral group:D8
Derived subgroup of M16M16Cyclic group:Z2Direct product of Z4 and Z2
Derived subgroup of nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Cyclic group:Z2Direct product of Z4 and Z2
First agemo subgroup of direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z2Klein four-group
Group of integers in group of rational numbersGroup of rational numbersGroup of integersGroup of rational numbers modulo integers
Klein four-subgroup of alternating group:A5Alternating group:A5Klein four-group
Klein four-subgroups of dihedral group:D8Dihedral group:D8Klein four-groupCyclic group:Z2
Non-central Z4 in M16M16Cyclic group:Z4Cyclic group:Z4
Non-characteristic order two subgroups of direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z2Cyclic group:Z4
Non-normal Klein four-subgroups of symmetric group:S4Symmetric group:S4Klein four-group
Non-normal subgroups of M16M16Cyclic group:Z2
Non-normal subgroups of dihedral group:D8Dihedral group:D8Cyclic group:Z2
Normal Klein four-subgroup of symmetric group:S4Symmetric group:S4Klein four-groupSymmetric group:S3
S2 in S3Symmetric group:S3Cyclic group:Z2
S2 in S4Symmetric group:S4Cyclic group:Z2
Subgroup generated by a non-commutator square in nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Cyclic group:Z2Dihedral group:D8
Subgroup generated by double transposition in symmetric group:S4Symmetric group:S4Cyclic group:Z2
Twisted S3 in A5Alternating group:A5Symmetric group:S3
Z2 in V4Klein four-groupCyclic group:Z2Cyclic group:Z2
Z4 in direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z4Cyclic group:Z2

## 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 Finite isomorph-free subgroup, Order-unique subgroup|FULL LIST, MORE INFO
normal Hall subgroup Hall subgroup (i.e., order and index are relatively prime) that is also normal Finite isomorph-free subgroup, Order-unique subgroup|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 Characteristic-isomorph-free subgroup, Injective endomorphism-invariant subgroup, Intermediately characteristic subgroup, Intermediately injective endomorphism-invariant subgroup, Isomorph-containing subgroup, Isomorph-normal characteristic subgroup, Normal-isomorph-free subgroup, Series-isomorph-free subgroup, Sub-(isomorph-normal characteristic) subgroup, Sub-isomorph-free subgroup|FULL LIST, MORE INFO
injective endomorphism-invariant subgroup invariant under all injective endomorphisms Intermediately injective endomorphism-invariant subgroup, Isomorph-containing subgroup|FULL LIST, MORE INFO
intermediately injective endomorphism-invariant subgroup injective endomorphism-invariant in all intermediate subgroups Isomorph-containing subgroup|FULL LIST, MORE INFO
intermediately characteristic subgroup characteristic in every intermediate subgroup (via isomorph-containing) (see also list of examples) Intermediately injective endomorphism-invariant subgroup, Isomorph-containing subgroup|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, Injective endomorphism-invariant subgroup, Intermediately characteristic subgroup, Isomorph-automorphic normal subgroup, Isomorph-containing subgroup, Isomorph-normal characteristic subgroup, Isomorph-normal subgroup, Sub-isomorph-free subgroup|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, Intermediately automorph-conjugate subgroup|FULL LIST, MORE INFO
intermediately automorph-conjugate subgroup automorph-conjugate in every intermediate subgroup Intermediately isomorph-conjugate 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 Normal-isomorph-free subgroup|FULL LIST, MORE INFO
series-isomorph-free subgroup normal and no isomorphic normal subgroup with isomorphic quotient Normal-isomorph-free subgroup|FULL LIST, MORE INFO
isomorph-normal subgroup every isomorphic subgroup is normal (see also list of examples) Isomorph-automorphic normal subgroup, Isomorph-characteristic subgroup, Isomorph-normal characteristic subgroup|FULL LIST, MORE INFO
isomorph-normal characteristic subgroup characteristic and every isomorphic subgroup is normal |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-freeness is not transitive It is possible to have groups $H \le K \le G$ such that $H$ is isomorph-free in $K$ and $K$ is isomorph-free in $G$ but $H$ is not isomorph-free in $G$.
transitive subgroup property Yes isomorph-freeness is quotient-transitive Suppose $H \le K \le G$ are groups such that $H$ is isomorph-free in $G$ and the quotient group $K/H$ is isomorph-free in $G/H$. Then, $K$ is isomorph-free in $G$.
intermediate subgroup condition Yes isomorph-freeness satisfies intermediate subgroup condition Suppose $H \le K \le G$ are groups such that $H$ is isomorph-free in $G$. Then, $H$ is also isomorph-free in $K$.
finite-intersection-closed subgroup property No isomorph-freeness is not finite-intersection-closed It is possible to have a group $G$ and subgroups $H, K$ of $G$ such that $H$ and $K$ are both isomorph-free but the intersection $H \cap K$ is not isomorph-free.
strongly join-closed subgroup property Yes isomorph-freeness is strongly join-closed Suppose $H_i, i \in I$ are subgroups of a group $G$ such that each $H_i$ is an isomorph-free subgroup of $G$. Then, the join of subgroups $\left \langle H_i \right \rangle_{i \in I}$ is also an isomorph-free subgroup of $G$.
finite-upper join-closed subgroup property No isomorph-freeness is not finite-upper join-closed It is possible to have groups $H \le G$ and $K,L$ are intermediate subgroups such that $H$ is isomorph-free in both $K$ and $L$, but $H$ is not isomorph-free in $\langle K, L \rangle$.
trivially true subgroup property Yes The trivial subgroup is isomorph-free in any group.
identity-true subgroup property No It is possible for a group to be isomorphic to a subgroup of itself. The simplest example is $\mathbb{Z}$, the group of integers, that is isomorphic to the subgroup $n\mathbb{Z}$ for any positive integer $n$. A group that is isomorph-free as a subgroup of itself is termed a co-Hopdian 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);