Definition:Subnormal Subgroup
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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 $i \in \left\{{0, 1, \ldots, k-1}\right\}$.
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.