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]

VIEW: Definitions built on this | Facts about this: (facts closely related to Hereditarily normal subgroup, all facts related to Hereditarily normal subgroup) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
View other semistandard definitions in group theory

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]

|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

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 (sometimes, quasicentral) if every subgroup of it is normal in the whole group.

Definition with symbols

A subgroup $H$ of a group $G$ is termed hereditarily normal (sometimes, quasicentral) if for any subgroup $K\leq H$, $K$ is normal in $G$.

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.

Relation with other properties

Stronger properties

• Central subgroup (subgroup lying inside the center)

Weaker properties

• SCAB-subgroup
• Transitively normal subgroup
• Hereditarily permutable subgroup
• Hereditarily subnormal subgroup
• Hereditarily pronormal subgroup
• Normal subgroup

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.