Intermediately automorph-conjugate implies intermediately characteristic in nilpotent

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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., Intermediately automorph-conjugate subgroup (?)) must also satisfy the second subgroup property (i.e., Intermediately characteristic subgroup (?)). In other words, every intermediately automorph-conjugate subgroup of nilpotent group is a intermediately characteristic subgroup of nilpotent group.
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Statement

In a nilpotent group, any intermediately automorph-conjugate subgroup is intermediately characteristic.

Related facts

Converse

The converse is clearly true in all groups: intermediately characteristic implies intermediately automorph-conjugate. In fact, in any group, characteristic implies automorph-conjugate.

Removing the intermediately operator

It is not in general true in a nilpotent group that every automorph-conjugate subgroup is characteristic. In fact, a group satisfying the property that every automorph-conjugate subgroup is characteristic is termed an ACIC-group. Some facts:

Facts used

  1. Intermediately automorph-conjugate implies intermediately normal-to-characteristic
  2. Intermediately normal-to-characteristic implies intermediately characteristic in nilpotent

Proof

The proof follows directly by combining facts (1) and (2).