Ascendant subgroup

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 every term in the series is a normal subgroup of its successor.

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

• Every subnormal subgroup is ascendant; every ascendant subgroup is serial.
• 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.