Additive group of a field implies monolith in holomorph

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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

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).