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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Contents
Statement
Propertytheoretic statement
For finite groups, the property of being ACIC is stronger than the property of being nilpotent.
Verbal statement
Any finite ACICgroup is nilpotent.
Definitions used
Automorphconjugate subgroup
Further information: automorphconjugate subgroup
A subgroup of a group is termed an automorphconjugate subgroup if for every automorphism of , and are conjugate subgroups.
ACICgroup
Further information: ACICgroup
A group is termed ACIC if every automorphconjugate subgroup is characteristic, or equivalently, any automorphconjugate subgroup is normal.
Finite nilpotent group
Further information: Finite nilpotent group
A finite group is termed nilpotent if all Sylow subgroups are normal (this is just one of the formulations of nilpotence for finite groups). The definition breaks down for infinite groups).
Facts used
 In a finite group, any Sylow subgroup is automorphconjugate. Further information: Sylow implies automorphconjugate
 The definition of finite nilpotent group given above: a finite nilpotent group is a finite group in which every Sylow subgroup is normal.
Related facts
Proof
Proof using subgroup property collapse
For any finite group, we have:
Sylow automorphconjugate
And for an ACICgroup, we have:
Automorphconjugate normal
Thus, for a finite ACICgroup, we have:
Sylow Normal
Which is precisely the condition for being a finite nilpotent group.