4-subnormal subgroup

Definition
A subgroup of a group is termed a 4-subnormal subgroup if it satisfies the following equivalent conditions:


 * It is a defining ingredient::subnormal subgroup of defining ingredient::subnormal depth at most four.
 * It is a defining ingredient::2-subnormal subgroup of a 2-subnormal subgroup.
 * It is a defining ingredient::normal subgroup of a defining ingredient::3-subnormal subgroup.
 * It is a 3-subnormal subgroup of a normal subgroup.

Stronger properties

 * Weaker than::Normal subgroup
 * Weaker than::2-subnormal subgroup
 * Weaker than::3-subnormal subgroup

Weaker properties

 * Stronger than::Subnormal subgroup

Metaproperties
A 4-subnormal subgroup of a 4-subnormal subgroup need not be 4-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 4-subnormal in $$G$$, $$H$$ is 4-subnormal in $$G$$. In fact, an analogous statement holds for all subnormal depths.

If $$H ,K \le G$$ with $$H$$ a 4-subnormal subgroup of $$G$$, $$H \cap K$$ is 4-subnormal in $$K$$. In fact, an analogous statement holds for all subnormal depths.

An arbitrary intersection of 4-subnormal subgroups is 4-subnormal. An analogous statement holds for all subnormal depths.

A join of two 4-subnormal subgroups need not be 4-subnormal; in fact, it need not even be subnormal. More generally, a join of 3-subnormal subgroups need not be subnormal.