# 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, $H$ is $k$ -subnormal in $G$ if there are subgroups

$H=H_{0},H_{1},H_{2},\ldots ,H_{k}=G$ of $G$ such that $H_{i}$ is normal in $H_{i+1}$ for each $i$ .

A subnormal subgroup is a subgroup that is $k$ -subnormal for some positive integer $k$ . 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.

## See also

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