Isomorph-free subgroup of finite group

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This article describes a property that arises as the conjunction of a subgroup property: isomorph-free subgroup with a group property imposed on the ambient group: finite group
View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup

Definition

An isomorph-free subgroup of finite group is a subgroup of a finite group satisfying the following equivalent conditions:

  1. It is an isomorph-free subgroup of the whole group: there is no other subgroup of the whole group isomorphic to it.
  2. It is an isomorph-containing subgroupisomorph-containing subgroup of the whole group: it contains every subgroup of the whole group isomorphic to it.

The equivalence of definitions follows from the fact that if a finite group contains an isomorphic subgroup, they are in fact equal.

Relation with other properties

Stronger properties

property quick description proof of implication proof of strictness (reverse implication failure) intermediate notions
Normal Sylow subgroup
Normal Hall subgroup
Order-contained subgroup
Prehomomorph-contained subgroup of finite group

Weaker properties

property quick description proof of implication proof of strictness (reverse implication failure) intermediate notions
Finite isomorph-free subgroup
Characteristic subgroup of finite group
Normal subgroup of finite group