Hereditarily subnormal subgroup

This article describes a property that arises as the conjunction of a subgroup property: subnormal subgroup with a group property (itself viewed as a subgroup property): group in which every subgroup is subnormal
View a complete list of such conjunctions

Definition

Symbol-free definition

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

1. It is subnormal in the whole group and is also a group in which every subgroup is subnormal: every subgroup of it is subnormal in it.
2. Every subgroup of the subgroup is subnormal in the whole group.

Equivalence of definitions

(1) implies (2) because subnormality is transitive, while (2) implies (1) because subnormality satisfies intermediate subgroup condition.

Relation with other properties

Stronger properties

• Hereditarily normal subgroup
• Hereditarily 2-subnormal subgroup
• Hereditarily fixed-depth subnormal subgroup: Also related:
  • Nilpotent subnormal subgroup
  • Abelian subnormal subgroup