Finite subgroup

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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 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