Additive group of a field implies monolith in holomorph

Statement
Suppose $$G$$ is isomorphic to the fact about::additive group of a field. Equivalently, $$G$$ is a fact about::characteristically simple group that is also an abelian group. In particular, $$G$$ is either an fact about::elementary abelian group or a direct sum of copies of the rationals.

Then, the holomorph of $$G$$ is a monolithic group with $$G$$ its monolith.

Analogues for other kinds of groups

 * Characteristically simple implies CSCFN-realizable (analogue for groups in general)
 * Characteristically simple and non-abelian implies monolith in automorphism group (analogue for non-abelian groups)

Other related facts

 * Characteristically simple and non-abelian implies automorphism group is complete
 * Semidirect product with self-normalizing subgroup of automorphism group of coprime order implies every automorphism is inner

Facts used

 * 1) uses::Abelian implies self-centralizing in holomorph
 * 2) uses::Every group is normal fully normalized in its holomorph
 * 3) uses::Characteristically simple and NSCFN implies monolith

Proof using given facts
The proof follows from facts (1)-(3).