Intermediately normal-to-characteristic implies intermediately characteristic in nilpotent

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 $H\leq K\leq G$, 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 $H\leq K\leq G$, $H$ 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 $H\leq K\leq G$, with $H$ a subnormal subgroup of $K$, $H$ is a normal subgroup of $K$.

Related facts

Corollaries

• Intermediately automorph-conjugate implies intermediately characteristic in nilpotent

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. $H\leq K\leq G$, $H$ 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.