Question:Characteristic subgroup definable

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This question has type complete description conjecture

Q: Is $H$ being a characteristic subgroup of $G$ equivalent to saying that there is a rule that allows us to determine $H$ uniquely in terms of $G$?

A: In a sense, yes, but it depends on what kind of "rule" you allow. Essentially, characteristic means invariant under all automorphisms, and therefore it means that there is a way of describing $H$ in terms of $G$ in an "isomorphism-invariant" or "group-theoretic" fashion.

So, this does imply that any subgroup-defining function, i.e., any rule that outputs a subgroup for every group in an isomorphism-invariant fashion, must always give a characteristic subgroup (see also subgroup-defining function value is characteristic). In particular, any subgroup that can be defined "uniquely" in terms of the whole group must be characteristic.

The converse is fuzzier and harder to formulate precisely. Whether or not every characteristic subgroup can be "uniquely defined" in terms of the whole group depends on how powerful a language we have at our disposal. If the language we have is arbitrarily powerful and allows for descriptions of infinite length, then yes, a subgroup is characteristic if and only if it can be "uniquely defined" in terms of the whole group.

However, if we put restrictions on the language we can use, there may be characteristic subgroups that cannot be defined uniquely. For instance, a purely definable subgroup is a subgroup that can be defined as a subset in the first-order theory of the pure group. Purely definable implies characteristic but characteristic not implies purely definable. In other words, it's possible that first-order language is not powerful enough to describe characteristic subgroups.