Almost normal subgroup

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

Symbol-free definition

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

Its normalizer has finite index in the whole group.
It is a normal subgroup of a subgroup of finite index in the whole group.
It has only finitely many conjugate subgroups.

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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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This is a variation of normal subgroup|Find other variations of normal subgroup | Read a survey article on varying normal subgroup

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Normal subgroup				\|FULL LIST, MORE INFO
Subgroup of finite index				\|FULL LIST, MORE INFO

Weaker properties

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

Related properties

Nearly normal subgroup

Facts

Every subgroup of a group is almost normal if and only if the center has finite index, or equivalently, if the inner automorphism group of the group is finite.

Metaproperties

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