Ascendant subgroup explained

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.

See also

• Descendant subgroup

References

• Book: Dixon, Martyn R.. Sylow Theory, Formations, and Fitting Classes in Locally Finite Groups . . 1994 . 981-02-1795-1 . 6 .
• Book: Robinson, Derek J.S. . A Course in the Theory of Groups . Derek J. S. Robinson. . 1996 . 0-387-94461-3 . 358 .