# 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
View other such compositions|View all subgroup properties

## Definition

### Symbol-free definition

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

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

This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]
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]
This subgroup property is a finitarily tautological subgroup property: when the ambient group is a finite group, the property is satisfied.
View other such subgroup properties
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

### 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

## 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 $H \le K \le G$ 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 $H \le K \le G$ and $H$ is almost normal in $G$, then $H$ is almost normal in $K$.
Transfer condition Yes Almost normal satisfies transfer condition If $H, K \le G$ 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 $H \le G$ 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 $H \le G$ 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.

## References

• Groups with finite classes of conjugate subgroups by B.H. Neumann, Math. Z., 63, 1955, Pages 76-96