Every nontrivial normal subgroup is potentially characteristic-and-not-intermediately characteristic

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Statement

Suppose G is a group and H is a nontrivial Normal subgroup (?) of G. Then, there exists a group K containing G such that H is a Characteristic subgroup (?) of K but not an intermediately characteristic subgroup of K (i.e., it is not characteristic in every intermediate subgroup).

Facts used

  1. Every nontrivial normal subgroup is potentially normal-and-not-characteristic
  2. NPC theorem: This statems that every normal subgroup is potentially characteristic.

Proof

Given: A group G, a nontrivial normal subgroup H of G.

To prove: There exists a group K containing G such that H is characteristic in K but not characteristic in every intermediate subgroup.

Proof:

  1. By fact (1), there exists a group L containing G such that H is normal and not characteristic in L.
  2. By fact (2), there exists a group K containing L such that H is characteristic in K.

Thus, H is characteristic in K but not in the intermediate subgroup L, completing the proof.