Potentially relatively characteristic equals normal

The result stated here is superseded by the following result, which is both stronger and simpler: normal equals potentially characteristic. In other words, the latter result has weaker and easier-to-verify hypotheses, and/or stronger and easier-to-use conclusions.
The main purpose of including this result is that it has a considerably easier proof, and/or was historically proved before the stronger result.

Statement

A subgroup of a group is termed potentially relatively characteristic if there is an embedding of the whole group in some bigger group such that every automorphism of the supergroup that restricts to an automorphism of the group, in fact restricts to an automorphism of the subgroup as well.

Then, a subgroup is normal if and only if it is potentially relatively characteristic.

Definition with symbols

A subgroup $H$ of a group $G$ is termed potentially relatively characteristic in $G$ if there is a group $K$ such that every automorphism of $K$ that restricts to an automorphism of $G$, also restricts to an automorphism of $H$.

$H$ is normal in $G$ if and only if $H$ is potentially relatively characteristic in $G$.

Related facts

• Normal equals potentially characteristic (a stronger statement requiring a more tricky construction)
• Extensible implies permutation-extensible (another statement superseded by a stronger statement, but interesting insofar as it uses a simple construction of a similar flavor)

Proof

Proof outline

The direction of potentially relatively characteristic implies normal is straightfoward.

For the other direction, the proof idea is as follows. Make the group $G$ act on the coset space of the normal subgroup $H$, and use this to get a homomorphism from the group to a symmetric group. A little trick can be used to get an injective homomorphism, and further, to ensure that the symmetric group is a complete group. Then, we show that inside this symmetric group, any automorphism that restricts to an automorphism of $G$ must also restrict to an automorphism of $H$.