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
amalgam-normal-subhomomorph-containing subgroup		finite normal implies amalgam-normal-subhomomorph-containing		|FULL LIST, MORE INFO
amalgam-characteristic subgroup	it is characteristic inside the join of the whole group with itself over the subgroup	finite normal implies amalgam-characteristic (also via amalgam-normal-subhomomorph-containing)	any central subgroup that is not finite works	Amalgam-normal-subhomomorph-containing subgroup, Amalgam-strictly characteristic subgroup, Periodic normal subgroup|FULL LIST, MORE INFO
quotient-powering-invariant subgroup	if the group is powered over a prime, so is the quotient group	finite normal implies quotient-powering-invariant	any infinite group as a subgroup of itself works	|FULL LIST, MORE INFO
powering-invariant subgroup	if the group is powered over a prime, so is the subgroup	(via quotient-powering-invariant)	(via quotient-powering-invariant)	Finite subgroup, Powering-invariant normal subgroup, Quotient-powering-invariant 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

Related properties

• Normal subgroup of finite index