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