# Hereditarily normal subgroup

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 article describes a property that arises as the conjunction of a subgroup property: transitively normal subgroup with a group property (itself viewed as a subgroup property): Dedekind group
View a complete list of such conjunctions

## Definition

### Symbol-free definition

A subgroup of a group is termed hereditarily normal or quasicentral if it satisfies the following equivalent conditions:

1. Every subgroup of the subgroup is a normal subgroup of the whole group.
2. It is both a transitively normal subgroup of the whole group and a Dedekind group.
3. All inner automorphisms of the whole group restrict to power automorphisms on the subgroup.

### Definition with symbols

A subgroup $H$ of a group $G$ is termed hereditarily normal or quasicentral if it satisfies the following equivalent conditions:

1. For every subgroup $K$ of $H$, $K$ is a normal subgroup of $G$.
2. $H$ is both a Dedekind group and a transitively normal subgroup of $G$.
3. For any $g \in G$, the inner automorphism of $G$ given by conjugation by $g$ restricts to a power automorphism of $H$.

## Formalisms

### In terms of the hereditarily operator

This property is obtained by applying the hereditarily operator to the property: normal subgroup
View other properties obtained by applying the hereditarily operator

The property of being hereditarily normal is a result of applying the hereditarily operator on the property of normality.

### Function restriction expression

This subgroup property is a function restriction-expressible subgroup property: it can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.
Find other function restriction-expressible subgroup properties | View the function restriction formalism chart for a graphic placement of this property
Function restriction expression $H$ is a hereditarily normal subgroup of $G$ if ... This means that "hereditarily normal" is ... Additional comments
inner automorphism $\to$ power automorphism every inner automorphism of $G$ restricts to a power automorphism of $H$
class-preserving automorphism $\to$ power automorphism every class-preserving automorphism of $G$ restricts to a power automorphism of $H$
subgroup-conjugating automorphism $\to$ power automorphism every subgroup-conjugating automorphism of $G$ restricts to a power automorphism of $H$
normal automorphism $\to$ power automorphism every normal automorphism of $G$ restricts to a power automorphism of $H$

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Central subgroup contained in the center central implies hereditarily normal hereditarily normal not implies central Abelian hereditarily normal subgroup|FULL LIST, MORE INFO
Cyclic normal subgroup normal and cyclic as a group Abelian hereditarily normal subgroup|FULL LIST, MORE INFO
Abelian hereditarily normal subgroup abelian subgroup and every subgroup of it is normal in whole group |FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
SCAB-subgroup every subgroup-conjugating automorphism of the whole group restricts to a subgroup-conjugating automorphism of the subgroup hereditarily normal implies SCAB SCAB not implies hereditarily normal |FULL LIST, MORE INFO
Transitively normal subgroup every normal subgroup of it is normal in the whole group SCAB-subgroup|FULL LIST, MORE INFO
Hereditarily permutable subgroup every subgroup of it is permutable in the whole group |FULL LIST, MORE INFO
Hereditarily subnormal subgroup every subgroup of it is subnormal in the whole group Nilpotent subnormal subgroup|FULL LIST, MORE INFO
Hereditarily pronormal subgroup every subgroup of it is pronormal in the whole group |FULL LIST, MORE INFO
Normal subgroup Dedekind normal subgroup, SCAB-subgroup, Transitively normal subgroup|FULL LIST, MORE INFO

## Metaproperties

### Left-hereditariness

This subgroup property is left-hereditary: any subgroup of a subgroup with this property also has this property. Hence, it is also a transitive subgroup property.

Since the left-hereditarily operator is idempotent, the property of being hereditarily normal is itself left hereditary (that is, every subgroup of a hereditarily normal subgroup is hereditarily normal).

Note that being a left-hereditary property, it is automatically transitive and also intersection-closed.

### Transfer condition

YES: This subgroup property satisfies the transfer condition: if a subgroup has the property in the whole group, its intersection with any subgroup has the property in that subgroup.
View other subgroup properties satisfying the transfer condition

Since normality satisfies the intermediate subgroup condition, and the left-hereditarily operator preserves the intermediate subgroup condition, the property of being hereditarily normal also satisfies the transfer. condition. Hence it also satisfies the intermediate subgroup condition.

### Trimness

The trivial group is obviously hereditarily normal.

The whole group is hereditarily normal if and only if every subgroup of the group is normal. Groups with this property are either Abelian groups or Hamiltonian groups.