Marginality does not satisfy intermediate subgroup condition

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This article gives the statement, and possibly proof, of a subgroup property (i.e., marginal subgroup) not satisfying a subgroup metaproperty (i.e., intermediate subgroup condition).
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Statement

It is possible to have a group G and subgroups H,K of G such that H \le K and:

Proof

Further information: dihedral group:D8, subgroup structure of dihedral group:D8

Consider the following:

Then, we have:

  • H is marginal in G, because center is marginal (it is marginal with respect to the variety of abelian groups).
  • H is not marginal in K, because it is not characteristic in K.