Category:Subnormal Subgroups

This category contains results about Subnormal Subgroups.

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 $G$.