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 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.
• Every subgroup can be expressed uniquely as an ascendant subgroup of a self-normalizing subgroup.

[edit] See also

• Descendant subgroup

[edit] References

• Martyn R. Dixon (1994). Sylow Theory, Formations, and Fitting Classes in Locally Finite Groups. World Scientific. p. 6. ISBN 9810217951. 
• Derek J.S. Robinson (1996). A Course in the Theory of Groups. Springer-Verlag. p. 358. ISBN 0-387-94461-3.