Isomorph-free subgroup

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 $K\le G$ such that $H\cong K$, then $H=K$ (i.e. $H$ and $K$ are the same subgroup).

Relation with other properties

Stronger properties

• Normal Sylow subgroup
• Normal Hall subgroup
• Order-unique subgroup

Weaker properties

• Intermediately characteristic subgroup
• Characteristic subgroup
• Normal subgroup
• Isomorph-conjugate subgroup
• Automorph-conjugate subgroup

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

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).