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.

See also[edit]

References[edit]

  • Martyn R. Dixon (1994). Sylow Theory, Formations, and Fitting Classes in Locally Finite Groups. World Scientific. p. 6. ISBN 981-02-1795-1. 
  • Derek J.S. Robinson (1996). A Course in the Theory of Groups. Springer-Verlag. p. 358. ISBN 0-387-94461-3.