# Finite normal subgroup

This article describes a property that arises as the conjunction of a subgroup property: normal 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 normal subgroup if it is finite (as a group) and normal as a subgroup.

## Examples

VIEW: subgroups satisfying this property | subgroups dissatisfying property normal subgroup | subgroups dissatisfying property finite group
VIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
normal subgroup of prime order normal subgroup and its order is a prime number
finite central subgroup finite and a central subgroup central implies normal any of the finite examples for normal not implies central |FULL LIST, MORE INFO
finite characteristic subgroup finite and a characteristic subgroup any of the finite examples for normal not implies characteristic |FULL LIST, MORE INFO
normal subgroup of finite group normal and the whole group is finite |FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
finitely generated normal subgroup normal subgroup and finitely generated as a group |FULL LIST, MORE INFO
normal closure of finite subset it is the normal subgroup generated by a finite subset of the whole group Finitely generated normal subgroup|FULL LIST, MORE INFO
finite subnormal subgroup finite and a subnormal subgroup |FULL LIST, MORE INFO
periodic normal subgroup normal subgroup in which every element has finite order |FULL LIST, MORE INFO
join-transitively finite subgroup its join with any finite subgroup of the whole group is finite any non-normal subgroup of a finite group works Finite elliptic subgroup|FULL LIST, MORE INFO
local powering-invariant subgroup if an element of the subgroup has a unique $n^{th}$ root in the group, that root is in the subgroup (via finite) Finite subgroup|FULL LIST, MORE INFO