3-subnormal subgroup

Symbol-free definition
A subgroup of a group is termed a 3-subnormal subgroup if it satisfies the following equivalent conditions:


 * 1) It is a subnormal subgroup and its subnormal depth is at most three.
 * 2) It is a 2-subnormal subgroup of a normal subgroup.
 * 3) It is a 2-subnormal subgroup in its normal closure.
 * 4) It is a normal subgroup of a 2-subnormal subgroup.

Stronger properties

 * Weaker than::Normal subgroup
 * Weaker than::2-subnormal subgroup
 * Weaker than::Commutator of a 2-subnormal subgroup and a subset:

Weaker properties

 * Stronger than::Subnormal subgroup
 * Stronger than::Conjugate-join-closed subnormal subgroup:

Metaproperties
A 3-subnormal subgroup of a 3-subnormal subgroup need not be 3-subnormal. This follows from the fact that there can be subgroups of arbitrarily large subnormal depth.

If $$H \le K \le G$$ and $$H$$ is 3-subnormal in $$G$$, $$H$$ is 3-subnormal in $$G$$. In fact, an analogous statement holds for all subnormal depths.

If $$H ,K \le G$$ with $$H$$ a 3-subnormal subgroup of $$G$$, $$H \cap K$$ is 3-subnormal in $$K$$. In fact, an analogous statement holds for all subnormal depths.

An arbitrary intersection of 3-subnormal subgroups is 3-subnormal. An analogous statement holds for all subnormal depths.

A join of two 3-subnormal subgroups need not be 3-subnormal; in fact, it need not even be subnormal.