Isomorph-free subgroup: Difference between revisions

Revision as of 22:23, 2 October 2008

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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Definition

Symbol-free definition

A subgroup of a group is said to be isomorph-free if there is no other subgroup of the group isomorphic to it as an abstract group.

Definition with symbols

A subgroup $H$ of a group $G$ is said to be isomorph-free if whenever $K \leq G$ such that $H ≅ K$ , then $H = K$ (i.e. $H$ and $K$ are the same subgroup).

Relation with other properties

Stronger properties

Weaker properties

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

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 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 $H \leq G$ 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).

@@ Line 48: / Line 48: @@
 {{proofat|[[Isomorph-freeness is join-closed]]}}
+{{not upper join-closed}}
+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]]}}
 ===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]]).