Subnormal depth

Definition
The subnormal depth (also sometimes called the defect or subnormal defect) of a defining ingredient::subnormal subgroup $$H$$ in a group $$G$$ is defined in the following equivalent ways:


 * It is the smallest $$n$$ for which there exists an ascending chain of subgroups $$H = H_0 \le H_1 \le H_2 \le \dots \le H_n = G$$ where each $$H_i$$ is normal in $$H_{i+1}$$.
 * Consider the sequence $$G_0 = G$$, $$G_i$$ is the normal closure of $$H$$ in $$G_{i-1}$$. The subnormal depth is the smallest $$n$$ for which $$G_n = H$$.
 * Consider the sequence $$K_i$$ where $$K_0 = G$$ and $$K_{i+1} = [H,K_i]$$. The subnormal depth is the smallest $$n$$ for which $$K_n \le H$$.

We typically say that a subgroup has subnormal depth $$k$$ if its subnormal depth is less than or equal to $$k$$. A subgroup of subnormal depth (less than or equal to) $$k$$ is termed a $$k$$-subnormal subgroup.

Particular cases

 * A subgroup has subnormal depth $$0$$ if and only if it is the whole group.
 * A subgroup has subnormal depth (less than or equal to) $$1$$ if and only if it is a normal subgroup.
 * A subgroup has subnormal depth (less than or equal to) $$2$$ if and only if it is a 2-subnormal subgroup.
 * A subgroup has subnormal depth (less than or equal to) $$3$$ if and only if it is a 3-subnormal subgroup.
 * A subgroup has subnormal depth (less than or equal to) $$4$$ if and only if it is a 4-subnormal subgroup.