Hereditarily normal subgroup

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

Metaproperties
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.

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.