Additive group of a field implies monolith in holomorph

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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 using given facts

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