# Isomorph-free subgroup of finite group

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: isomorph-free subgroup with a group property imposed on theambient 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

## Contents

## 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 |