ACIC implies nilpotent (finite groups)
This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property must also satisfy the second group property
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Further information: automorph-conjugate subgroup
A subgroup of a group is termed an automorph-conjugate subgroup if for every automorphism of , and are conjugate subgroups.
Further information: ACIC-group
Finite nilpotent group
Further information: Finite nilpotent group
- In a finite group, any Sylow subgroup is automorph-conjugate. Further information: Sylow implies automorph-conjugate
- The definition of finite nilpotent group given above: a finite nilpotent group is a finite group in which every Sylow subgroup is normal.
Proof using subgroup property collapse
For any finite group, we have:
And for an ACIC-group, we have:
Thus, for a finite ACIC-group, we have:
Which is precisely the condition for being a finite nilpotent group.