# Finite subgroup

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: subgroup with a group property (itself viewed as a subgroup property): finite group

View a complete list of such conjunctions

## Contents

## Definition

A subgroup of a group is termed a **finite subgroup** if, as a group by itself, it is a finite group.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Finite characteristic subgroup | ||||

Finite normal subgroup | ||||

Join-transitively finite subgroup | ||||

Finite elliptic subgroup | ||||

Subgroup of finite group |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

finitely generated subgroup | |FULL LIST, MORE INFO | |||

local powering-invariant subgroup | finite implies local powering-invariant | any infinite group as a subgroup of itself | Intermediately local powering-invariant subgroup, Periodic subgroup|FULL LIST, MORE INFO | |

powering-invariant subgroup | finite implies powering-invariant | any infinite group as a subgroup of itself | Intermediately local powering-invariant subgroup, Intermediately powering-invariant subgroup, LCS-powering-invariant subgroup, Local powering-invariant subgroup, Periodic subgroup|FULL LIST, MORE INFO |