Additive group of a field implies monolith in holomorph

Statement

Suppose $G$ is isomorphic to the Additive group of a field (?). Equivalently, $G$ is a Characteristically simple group (?) that is also an abelian group. In particular, $G$ is either an 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.

Related facts

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. Abelian implies self-centralizing in holomorph
2. Every group is normal fully normalized in its holomorph
3. Characteristically simple and NSCFN implies monolith

Proof

Proof using given facts

The proof follows from facts (1)-(3).