Definition:Subnormal Subgroup

Definition
Let $G$ be a group.

Let $H$ be a subgroup of $G$.

Then $H$ is a $k$-subnormal subgroup of $G$ iff there is a finite sequence of subgroups of $G$:
 * $H = H_0, H_1, H_2, \ldots, H_k = G$

such that $H_i$ is a normal subgroup of $H_{i+1}$ for all $i \in \left\{{0, 1, \ldots, k-1}\right\}$.

That is, there exists a normal series:
 * $H = H_0 \triangleleft H_1 \triangleleft H_2 \triangleleft \cdots \triangleleft H_k = G$

where $\triangleleft$ denotes the relation of normality.

Thus $H$ is a subnormal subgroup of $G$ if there exists $k \in \Z_{>0}$ such that $H$ is a $k$-subnormal subgroup of $G$.

By this definition, a normal subgroup is a $1$-subnormal subgroup.

Also see

 * Definition:Normal Subgroup
 * Definition:Abnormal Subgroup
 * Definition:Weakly Abnormal Subgroup
 * Definition:Contranormal Subgroup
 * Definition:Self-Normalizing Subgroup
 * Definition:Pronormal Subgroup
 * Definition:Weakly Pronormal Subgroup
 * Definition:Paranormal Subgroup
 * Definition:Polynormal Subgroup