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]

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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 part	Subgroup part	Quotient part
A3 in S3	Symmetric group:S3	Cyclic group:Z3	Cyclic group:Z2
A4 in S4	Symmetric group:S4	Alternating group:A4	Cyclic group:Z2
Center of nontrivial semidirect product of Z4 and Z4	Nontrivial semidirect product of Z4 and Z4	Klein four-group	Klein four-group
Center of quaternion group	Quaternion group	Cyclic group:Z2	Klein four-group
Cyclic maximal subgroup of dihedral group:D16	Dihedral group:D16	Cyclic group:Z8	Cyclic group:Z2
Cyclic maximal subgroup of dihedral group:D8	Dihedral group:D8	Cyclic group:Z4	Cyclic group:Z2
Cyclic maximal subgroup of semidihedral group:SD16	Semidihedral group:SD16	Cyclic group:Z8	Cyclic group:Z2
D8 in SD16	Semidihedral group:SD16	Dihedral group:D8	Cyclic group:Z2
Derived subgroup of dihedral group:D16	Dihedral group:D16	Cyclic group:Z4	Klein four-group
Direct product of Z4 and Z2 in M16	M16	Direct product of Z4 and Z2	Cyclic group:Z2
First omega subgroup of direct product of Z4 and Z2	Direct product of Z4 and Z2	Klein four-group	Cyclic group:Z2
Klein four-subgroup of M16	M16	Klein four-group	Cyclic group:Z4
Klein four-subgroup of alternating group:A4	Alternating group:A4	Klein four-group	Cyclic group:Z3
Q8 in SD16	Semidihedral group:SD16	Quaternion group	Cyclic group:Z2
Q8 in central product of D8 and Z4	Central product of D8 and Z4	Quaternion group	Cyclic 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 part	Subgroup part	Quotient part
Center of M16	M16	Cyclic group:Z4	Klein four-group
Center of dihedral group:D16	Dihedral group:D16	Cyclic group:Z2	Dihedral group:D8
Center of dihedral group:D8	Dihedral group:D8	Cyclic group:Z2	Klein four-group
Center of semidihedral group:SD16	Semidihedral group:SD16	Cyclic group:Z2	Dihedral group:D8
Center of unitriangular matrix group:UT(3,p)	Unitriangular matrix group:UT(3,p)	Group of prime order	Elementary abelian group of prime-square order
Cyclic maximal subgroups of quaternion group	Quaternion group	Cyclic group:Z4	Cyclic group:Z2
D8 in D16	Dihedral group:D16	Dihedral group:D8	Cyclic group:Z2
Derived subgroup of M16	M16	Cyclic group:Z2	Direct product of Z4 and Z2
Derived subgroup of nontrivial semidirect product of Z4 and Z4	Nontrivial semidirect product of Z4 and Z4	Cyclic group:Z2	Direct product of Z4 and Z2
First agemo subgroup of direct product of Z4 and Z2	Direct product of Z4 and Z2	Cyclic group:Z2	Klein four-group
Group of integers in group of rational numbers	Group of rational numbers	Group of integers	Group of rational numbers modulo integers
Klein four-subgroups of dihedral group:D8	Dihedral group:D8	Klein four-group	Cyclic group:Z2
Non-central Z4 in M16	M16	Cyclic group:Z4	Cyclic group:Z4
Non-characteristic order two subgroups of direct product of Z4 and Z2	Direct product of Z4 and Z2	Cyclic group:Z2	Cyclic group:Z4
Normal Klein four-subgroup of symmetric group:S4	Symmetric group:S4	Klein four-group	Symmetric group:S3
S2 in S3	Symmetric group:S3	Cyclic group:Z2
Subgroup generated by a non-commutator square in nontrivial semidirect product of Z4 and Z4	Nontrivial semidirect product of Z4 and Z4	Cyclic group:Z2	Dihedral group:D8
Z2 in V4	Klein four-group	Cyclic group:Z2	Cyclic group:Z2
Z4 in direct product of Z4 and Z2	Direct product of Z4 and Z2	Cyclic group:Z4	Cyclic 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
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	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
isomorph-conjugate subgroup	all isomorphic subgroups are conjugate to it	obvious	(see also list of examples)	Intermediately isomorph-conjugate 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-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

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);