Nearly normal subgroup

This page describes a subgroup property obtained as a composition of two fundamental subgroup properties: subgroup of finite index and normal subgroup
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Definition

Symbol-free definition

A subgroup of a group is said to be nearly normal if it satisfies the following equivalent conditions:

It has finite index in its normal closure.
It is a subgroup of finite index of a normal subgroup of the whole group.

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties [SHOW MORE]
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This subgroup property is a finitarily tautological subgroup property: when the ambient group is a finite group, the property is satisfied.
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Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Normal subgroup
Subgroup of finite index

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Conjugate-commensurable subgroup
Subgroup for which any join of conjugates is a join of finitely many conjugates
Almost subnormal subgroup

Related properties

Almost normal subgroup

Facts

Every subgroup of a group is nearly normal if and only if the derived subgroup is finite, viz., the group is a commutator-finite group.

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
Transitive subgroup property	No		It is possible to have $H \le K \le G$ with $H$ nearly normal in $K$ and $K$ nearly normal in $G$ , but $H$ is not nearly normal in $G$ .
Trim subgroup property	Yes		Every group is nearly normal in itself; the trivial subgroup is nearly normal in every group.
Intermediate subgroup condition	Yes	Nearly normal satisfies intermediate subgroup condition	If $H \le K \le G$ and $H$ is nearly normal in $G$ , then $H$ is nearly normal in $K$ .
Transfer condition	Yes	Nearly normal satisfies transfer condition	If $H, K \le G$ with $H$ nearly normal in $G$ , then $H \cap K$ is nearly normal in $K$ .
Inverse image condition	Yes	Nearly normal satisfies inverse image condition	If $H \le G$ is nearly normal and $\varphi:M \to G$ is a homomorphism, then $\varphi^{-1}(H)$ is nearly normal in $M$ .
Image condition	Yes	Nearly normal satisfies image condition	If $H \le G$ is nearly normal and $\varphi:G \to M$ is a surjective homomorphism, then $\varphi(H)$ is nearly normal in $M$ .
Finite-intersection-closed subgroup property	Yes	Nearly normal is finite-intersection-closed	If $H,K$ are nearly normal subgroups of $G$ , then $H \cap K$ is also a nearly normal subgroup.
Finite-join-closed subgroup property	Yes	Nearly normal is finite-join-closed	If $H,K$ are nearly normal subgroups of $G$ , then $\langle H, K \rangle$ is also a nearly normal subgroup.
Conjugate-join-closed subgroup property	Yes	Nearly normal is conjugate-join-closed	A join of any number of conjugates of a nearly normal subgroup of a group is nearly normal.