Nilpotent implies every normal subgroup is potentially characteristic
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a nilpotent group. That is, it states that in a Nilpotent group (?), every subgroup satisfying the first subgroup property (i.e., Normal subgroup (?)) must also satisfy the second subgroup property (i.e., Potentially characteristic subgroup (?)). In other words, every normal subgroup of nilpotent group is a potentially characteristic subgroup of nilpotent group.
View all subgroup property implications in nilpotent groups View all subgroup property non-implications in nilpotent groups View all subgroup property implications View all subgroup property non-implications
This fact is related to: NPC conjecture
View other facts related to NPC conjectureView terms related to NPC conjecture |
- Finite normal implies potentially characteristic
- Central implies potentially characteristic
- Finite implies every normal subgroup is potentially characteristic
- Abelian implies every subgroup is potentially characteristic
The proof follows from fact (1), and the observation that the hypercenter of a nilpotent group equals the whole group.