# 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
View other such compositions|View all subgroup properties

## Definition

### Symbol-free definition

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

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

## 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.

## References

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