Hereditarily normal subgroup

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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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 $G$.

In terms of the left hereditary operator

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

Relation with other properties

Stronger properties

• Central subgroup (subgroup lying inside the center)

Weaker properties

• Transitively normal subgroup
• Hereditarily permutable 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.