Normal not implies normal-potentially relatively characteristic

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This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., normal subgroup) need not satisfy the second subgroup property (i.e., normal-potentially relatively characteristic subgroup)
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Statement

A normal subgroup of a group need not be a normal-potentially relatively characteristic subgroup.

Facts used

  1. Normal not implies normal-extensible automorphism-invariant
  2. Normal-potentially relatively characteristic

Proof

The proof follows directly from facts (1) and (2).

Example of the dihedral group

Further information: dihedral group:D8

Let G be the dihedral group of order eight, and H be one of the Klein four-subgroups.