# Subnormal subgroup

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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, is -subnormal in if there are subgroups

of such that is normal in for each .

A subnormal subgroup is a subgroup that is -subnormal for some positive integer . Some facts about subnormal subgroups:

- A 1-subnormal subgroup is a proper normal subgroup (and vice versa).
- A finitely generated group is nilpotent if and only if each of its subgroups is subnormal.
- Every quasinormal subgroup, and, more generally, every conjugate permutable subgroup, of a finite group is subnormal.
- Every pronormal subgroup that is also subnormal, is, in fact, normal. In particular, a Sylow subgroup is subnormal if and only if it is normal.
- Every 2-subnormal subgroup is a conjugate permutable subgroup.

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

## See also

- Normal subgroup
- Characteristic subgroup
- Normal core
- Normal closure
- Ascendant subgroup
- Descendant subgroup
- Serial subgroup

## References

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