Hereditarily subnormal subgroup

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.

Stronger properties

 * Weaker than::Hereditarily normal subgroup
 * Weaker than::Hereditarily 2-subnormal subgroup
 * Weaker than::Hereditarily fixed-depth subnormal subgroup: Also related:
 * Weaker than::Nilpotent subnormal subgroup
 * Weaker than::Abelian subnormal subgroup