Definition:Subnormal Subgroup

This page is about subnormal subgroup. For other uses, see subnormal.

Definition

Let $G$ be a group.

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

Then $H$ is a $k$-subnormal subgroup of $G$ if and only if 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 $i \in \set {0, 1, \ldots, k - 1}$.

That is, if and only if there exists a normal series:

$H = H_0 \lhd H_1 \lhd H_2 \lhd \cdots \lhd H_k = G$

where $\lhd$ denotes the relation of normality.

Thus $H$ is a subnormal subgroup of $G$ if and only 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

• Results about subnormal subgroups can be found here.

Sources

There are no source works cited for this page. Source citations are highly desirable, and mandatory for all definition pages. Definition pages whose content is wholly or partly unsourced are in danger of having such content deleted. To discuss this page in more detail, feel free to use the talk page.