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ascendent$5197$ - translation to greek
TYPE OF SUBGROUP IN GROUP THEORY
Ascendent subgroup
ascendent
adj. κυρίαρχος, επικρατών
Wikipedia
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.
