# Subnormal subgroup

In mathematics, in the field of group theory, a subgroup H of a given group G is a subnormal subgroup of G if there is a finite chain of subgroups of the group, each one normal in the next, beginning at H and ending at G.

In notation, ${\displaystyle H}$ is ${\displaystyle k}$-subnormal in ${\displaystyle G}$ if there are subgroups

${\displaystyle H=H_{0},H_{1},H_{2},\ldots ,H_{k}=G}$

of ${\displaystyle G}$ such that ${\displaystyle H_{i}}$ is normal in ${\displaystyle H_{i+1}}$ for each ${\displaystyle i}$.

A subnormal subgroup is a subgroup that is ${\displaystyle k}$-subnormal for some positive integer ${\displaystyle k}$. Some facts about subnormal subgroups:

The property of subnormality is transitive, that is, a subnormal subgroup of a subnormal subgroup is subnormal. The relation of subnormality can be defined as the transitive closure of the relation of normality.

If every subnormal subgroup of G is normal in G, then G is called a T-group.