Isomorph-free subgroup of finite group
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:
- It is an isomorph-free subgroup of the whole group: there is no other subgroup of the whole group isomorphic to it.
- 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 |