Isomorph-free subgroup: Difference between revisions

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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:

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

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.

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

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	Statement with symbols
transitive subgroup property	No	isomorph-freeness is not transitive	It is possible to have groups $H \leq K \leq G$ 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 $H \leq K \leq G$ 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 $H \leq K \leq G$ 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 ${⟨ H_{i} ⟩}_{i \in I}$ 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 $H \leq G$ and $K, L$ are intermediate subgroups such that $H$ is isomorph-free in both $K$ and $L$ , but $H$ is not isomorph-free in $⟨ K, L ⟩$ .
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 $Z$ , the group of integers, that is isomorphic to the subgroup $n Z$ 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 $H = H_{0} \leq H_{1} \leq \dots \leq H_{n} = G$ , 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);

@@ Line 1: / Line 1: @@
-{{subgroup property}}
+{{wikilocal}}
+==Definition==
+{{quick phrase|[[quick phrase::no other isomorphic subgroups]], [[quick phrase::no isomorphic copies]], [[quick phrase::only subgroup of its isomorphism type]]}}
+A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is said to be '''isomorph-free''' if it satisfies the following equivalent conditions:
-{{variationof|characteristicity}}
+# Whenever <math>K \le G</math> such that <math>H \cong K</math>, then <math>H = K</math> (i.e. <math>H</math> and <math>K</math> are the ''same'' subgroup).
+# <math>H</math> is a [[co-Hopfian group]], and whenever <math>K \le G</math> such that <math>H \cong K</math>, then <math>K \le H</math>.
-{{wikilocal}}
+{{subgroup property}}
+{{variation of|characteristic subgroup}}
+{{group-subgroup property conjunction|isomorph-containing subgroup|co-Hopfian group}}
-==Definition==
+==Examples==
-===Symbol-free definition===
+===Extreme examples===
-A [[subgroup]] of a [[group]] is said to be '''isomorph-free''' if there is no other subgroup of the group [[isomorphic groups|isomorphic]] to it as an abstract group.
+* The trivial subgroup is isomorph-free.
+* Any [[co-Hopfian group]] (and in particular, any [[finite group]]) is isomorph-free as a subgroup of itself.
-===Definition with symbols===
+===Examples in small finite groups===
-A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is said to be '''isomorph-free''' if whenever <math>K \le G</math> such that <math>H \cong K</math>, then <math>H = K</math> (i.e. <math>H</math> and <math>K</math> are the ''same'' subgroup).
+{{subgroup property see examples embed|isomorph-free subgroup}}
 ==Relation with other properties==
@@ Line 19: / Line 27: @@
 ===Stronger properties===
-* [[Weaker than::Normal Sylow subgroup]]
+{| class="sortable" border="1"
-* [[Weaker than::Normal Hall subgroup]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Weaker than::Order-unique subgroup]]
+|-
+| [[Weaker than::normal Sylow subgroup]] || [[Sylow subgroup]] that is also normal || Follows from [[Sylow implies order-conjugate]] || || {{intermediate notions short|isomorph-free subgroup|normal Sylow subgroup}}
+|-
+| [[Weaker than::normal Hall subgroup]] || [[Hall subgroup]] (i.e., order and index are relatively prime) that is also normal || || || {{intermediate notions short|isomorph-free subgroup|normal Hall subgroup}}
+|-
+| [[Weaker than::order-unique subgroup]] || unique subgroup of its order || || [[isomorph-free not implies order-unique]] {{strictness examples for subgroup property|isomorph-free subgroup|order-unique subgroup}} || {{intermediate notions short|isomorph-free subgroup|order-unique subgroup}}
+|}
 ===Weaker properties===
-* [[Stronger than::Intermediately characteristic subgroup]]
+{| class="sortable" border="1"
-* [[Stronger than::Characteristic subgroup]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Stronger than::Normal subgroup]]
+|-
-* [[Stronger than::Isomorph-conjugate subgroup]]
+| [[Stronger than::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]] ||
-* [[Stronger than::Intermediately isomorph-conjugate subgroup]]
+|-
-* [[Stronger than::Automorph-conjugate subgroup]]
+| [[Stronger than::characteristic subgroup]] || invariant under all [[automorphism]]s || ([[isomorph-containing implies characteristic|via isomorph-containing]]) || [[characteristic not implies isomorph-free in finite]] || {{intermediate notions short|characteristic subgroup|isomorph-free subgroup}}
-* [[Stronger than::Intermediately automorph-conjugate subgroup]]
+|-
-* [[Stronger than::Normal-isomorph-free subgroup]]
+| [[Stronger than::injective endomorphism-invariant subgroup]] ||invariant under all [[injective endomorphism]]s || || || {{intermediate notions short|injective endomorphism-invariant subgroup|isomorph-free subgroup}}
-* [[Stronger than::Characteristic-isomorph-free subgroup]]
+|-
+| [[Stronger than::intermediately injective endomorphism-invariant subgroup]] || injective endomorphism-invariant in all intermediate subgroups || || || {{intermediate notions short|intermediately injective endomorphism-invariant subgroup|isomorph-free subgroup}}
+|-
+| [[Stronger than::intermediately characteristic subgroup]] || characteristic in every intermediate subgroup || || ([[intermediately characteristic not implies isomorph-containing in abelian group|via isomorph-containing]]) {{strictness examples for subgroup property|intermediately characteristic subgroup|isomorph-free subgroup}} || {{intermediate notions short|intermediately characteristic subgroup|isomorph-free subgroup}}
+|-
+| [[Stronger than::normal subgroup]] || invariant under all [[inner automorphism]]s || ([[characteristic implies normal|via characteristic]]) || ([[normal not implies characteristic|via characteristic]]) {{strictness examples for subgroup property|normal subgroup|isomorph-free subgroup}} || {{intermediate notions short|normal subgroup|isomorph-free subgroup}}
+|-
+| [[Stronger than::isomorph-conjugate subgroup]] || all isomorphic subgroups are [[conjugate subgroups|conjugate]] to it || obvious || {{strictness examples for subgroup property|isomorph-conjugate subgroup|isomorph-free subgroup}} || {{intermediate notions short|isomorph-conjugate subgroup|isomorph-free subgroup}}
+|-
+| [[Stronger than::intermediately isomorph-conjugate subgroup]] || isomorph-conjugate in every intermediate subgroup || || || {{intermediate notions short|intermediately isomorph-conjugate subgroup|isomorph-free subgroup}}
+|-
+| [[Stronger than::automorph-conjugate subgroup]] || all [[automorphic subgroups]] are [[conjugate subgroups|conjugate]] to it || (via isomorph-conjugate) || (via isomorph-conjugate) || {{intermediate notions short|automorph-conjugate subgroup|isomorph-free subgroup}}
+|-
+| [[Stronger than::intermediately automorph-conjugate subgroup]] || automorph-conjugate in every intermediate subgroup || || || {{intermediate notions short|intermediately automorph-conjugate subgroup|isomorph-free subgroup}}
+|-
+| [[Stronger than::normal-isomorph-free subgroup]] || [[normal subgroup|normal]] and no other isomorphic normal subgroup || || || {{intermediate notions short|normal-isomorph-free subgroup|isomorph-free subgroup}}
+|-
+| [[Stronger than::characteristic-isomorph-free subgroup]] || [[characteristic subgroup|characteristic]] and no other isomorphic characteristic subgroup || || || {{intermediate notions short|characteristic-isomorph-free subgroup|isomorph-free subgroup}}
+|-
+| [[Stronger than::series-isomorph-free subgroup]] || [[normal subgroup|normal]] and no isomorphic normal subgroup with isomorphic quotient || || || {{intermediate notions short|series-isomorph-free subgroup|isomorph-free subgroup}}
+|-
+| [[Stronger than::isomorph-normal subgroup]] || every isomorphic subgroup is normal || || {{strictness examples for subgroup property|isomorph-normal subgroup|isomorph-free subgroup}} || {{intermediate notions short|isomorph-normal subgroup|isomorph-free subgroup}}
+|-
+| [[Stronger than::isomorph-characteristic subgroup]] || every isomorphic subgroup is characteristic || || {{strictness examples for subgroup property|isomorph-characteristic subgroup|isomorph-free subgroup}} || {{intermediate notions short|isomorph-characteristic subgroup|isomorph-free subgroup}}
+|-
+| [[Stronger than::isomorph-normal characteristic subgroup]] || characteristic and every isomorphic subgroup is normal || || || {{intermediate notions short|isomorph-normal characteristic subgroup|isomorph-free subgroup}}
+|}
 ==Metaproperties==
-{{intransitive}}
+{{wikilocal-section}}
-An isomorph-free subgroup of an isomorph-free subgroup need not be isomorph-free. {{further|[[Isomorph-freeness is not transitive]]}}
-{{quot-transitive}}
-If <math>H</math> is an isomorph-free subgroup of <math>G</math> and <math>K/H</math> is an isomorph-free subgroup of <math>G/H</math>, then <math>K</math> is an isomorph-free subgroup of <math>G</math>.
-{{proofat|[[Isomorph-freeness is quotient-transitive]]}}
-{{not intersection-closed}}
-If <math>H, K</math> are isomorph-free subgroups of <math>G</math>, the intersection <math>H \cap K</math> need not be isomorph-free. {{proofat|[[Isomorph-freeness is not intersection-closed]]}}
-{{join-closed}}
-If <math>H_i, i \in I</math> is a collection of isomorph-free subgroups of <math>G</math>, the join of the <math>H_i</math>s is also isomorph-free.
-{{proofat|[[Isomorph-freeness is strongly join-closed]]}}
+Here is a summary:
-{{not upper join-closed}}
+{| class="sortable" border="1"
+!Metaproperty name !! Satisfied? !! Proof !! Difficulty level (0-5) !! Statement with symbols
+|-
+| [[dissatisfies metaproperty::transitive subgroup property]] || No || [[isomorph-freeness is not transitive]] || {{#show: isomorph-freeness is not transitive | ?Difficulty level}} || It is possible to have groups <math>H \le K \le G</math> such that <math>H</math> is isomorph-free in <math>K</math> and <math>K</math> is isomorph-free in <math>G</math> but <math>H</math> is not isomorph-free in <math>G</math>.
+|-
+| [[satisfies metaproperty::transitive subgroup property]] || Yes || [[isomorph-freeness is quotient-transitive]] || {{#show: isomorph-freeness is quotient-transitive | ?Difficulty level}} || Suppose <math>H \le K \le G</math> are groups such that <math>H</math> is isomorph-free in <math>G</math> and the [[quotient group]] <math>K/H</math> is isomorph-free in <math>G/H</math>. Then, <math>K</math> is isomorph-free in <math>G</math>.
+|-
+| [[satisfies metaproperty::intermediate subgroup condition]] || Yes || [[isomorph-freeness satisfies intermediate subgroup condition]] || {{#show: isomorph-freeness satisfies intermediate subgroup condition | ?Difficulty level}}|| Suppose <math>H \le K \le G</math> are groups such that <math>H</math> is isomorph-free in <math>G</math>. Then, <math>H</math> is also isomorph-free in <math>K</math>.
+|-
+| [[dissatisfies metaproperty::finite-intersection-closed subgroup property]] || No || [[isomorph-freeness is not finite-intersection-closed]] || {{#show: isomorph-freeness is not finite-intersection-closed | ?Difficulty level}}|| It is possible to have a group <math>G</math> and subgroups <math>H, K</math> of <math>G</math> such that <math>H</math> and <math>K</math> are both isomorph-free but the [[intersection of subgroups|intersection]] <math>H \cap K</math> is not isomorph-free.
+|-
+| [[satisfies metaproperty::strongly join-closed subgroup property]] || Yes || [[isomorph-freeness is strongly join-closed]] || {{#show: isomorph-freeness is strongly join-closed | ?Difficulty level}} || Suppose <math>H_i, i \in I</math> are subgroups of a group <math>G</math> such that each <math>H_i</math> is an isomorph-free subgroup of <math>G</math>. Then, the [[join of subgroups]] <math>\left \langle H_i \right \rangle_{i \in I}</math> is also an isomorph-free subgroup of <math>G</math>.
+|-
+| [[dissatisfies metaproperty::finite-upper join-closed subgroup property]] || No || [[isomorph-freeness is not finite-upper join-closed]] || {{#show: isomorph-freeness is finite-upper join-closed | ?Difficulty level}} || It is possible to have groups <math>H \le G</math> and <math>K,L</math> are intermediate subgroups such that <math>H</math> is isomorph-free in both <math>K</math> and <math>L</math>, but <math>H</math> is not isomorph-free in <math>\langle K, L \rangle</math>.
+|-
+| [[satisfies metaproperty::trivially true subgroup property]] || Yes || || || The trivial subgroup is isomorph-free in any group.
+|-
+| [[dissatisfies metaproperty::identity-true subgroup property]] || No || || || It is possible for a group to be isomorphic to a subgroup of itself. The simplest example is <math>\mathbb{Z}</math>, the [[group of integers]], that is isomorphic to the subgroup <math>n\mathbb{Z}</math> for any positive integer <math>n</math>. A group that is isomorph-free as a subgroup of itself is termed a [[co-Hopdian group]].
+|}
-If <math>H \le G</math> and <math>K,L</math> are intermediate subgroups such that <math>H</math> is isomorph-free in both <math>K</math> and <math>L</math>, <math>H</math> need not be isomorph-free in <math>\langle K, L \rangle</math>. {{proofat|[[Isomorph-freeness is not upper join-closed]]}}
+==Effect of property operators==
-===Trimness===
+{{applyingoperatorgives|subordination operator|sub-isomorph-free subgroup}}
-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]]).
+A subgroup <math>H</math> of a group <math>G</math> is termed '''sub-isomorph-free''' if there is a series of subgroups <math>H = H_0 \le H_1 \le \dots \le H_n = G</math>, with each <math>H_{i-1}</math> an [[isomorph-free subgroup]] of <math>H_i</math>.
 ==Testing==