Ascendant subgroup

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In mathematics, in the field of group theory, a subgroup of a group is said to be ascendant if there is an ascending series starting from the subgroup and ending at the group, such that evey term in the series is a normal subgroup of its successor.

The series may be infinite. If the series if finite, then the subgroup is subnormal. Here are some properties of ascendant subgroups:

Every subnormal subgroup is ascendant.
In a finite group, the properties of being ascendant and subnormal are equivalent.
An arbitrary intersection of ascendant subgroups is ascendant.
Given any subgroup, there is a minimal ascendant subgroup containing it.
Every subgroup can be expressed uniquely as an ascendant subgroup of a self-normalizing subgroup.