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

In a nilpotent group, any intermediately normal-to-characteristic subgroup is an intermediately characteristic subgroup.

Definitions used

Intermediately normal-to-characteristic subgroup

Further information: Intermediately normal-to-characteristic subgroup

A subgroup H of a group G is termed intermediately normal-to-characteristic in G if whenever H \le K \le G, with H a normal subgroup of K, H is a characteristic subgroup of K.

Intermediately characteristic subgroup

Further information: Intermediately characteristic subgroup

A subgroup H of a group G is termed intermediately characteristic in G if whenever H \le K \le G, H is a characteristic subgroup of K.

Intermediately subnormal-to-normal subgroup

Further information: Intermediately subnormal-to-normal subgroup

A subgroup H of a group G is termed intermediately subnormal-to-normal in G if whenever H \le K \le G, with H a subnormal subgroup of K, H is a normal subgroup of K.

Related facts

Corollaries

Facts used

  1. Intermediately normal-to-characteristic implies intermediately subnormal-to-normal
  2. Nilpotence is subgroup-closed
  3. Nilpotent implies every subgroup is subnormal

Proof

Given: A nilpotent group G, an intermediately normal-to-characteristic subgroup H of G.

To prove: Whenever K is an intermediate subgroup, i.e. H \le K \le G, H is characteristic in K.

Proof: By fact (2), K is nilpotent and by fact (3), H is a subnormal subgroup of K.

By fact (1), H is intermediately subnormal-to-normal in G, and combinig this with the fact that H is subnormal in K yields that H is normal in K. Combining this with the given fact that H is intermediately normal-to-characteristic in G yields that H is characteristic in K, completing the proof.