Additive group of a field implies characteristic in holomorph

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This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., additive group of a field) must also satisfy the second group property (i.e., holomorph-characteristic group)
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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, G is a Characteristic subgroup (?) in its holomorph.

Related facts

Facts used

  1. Additive group of a field implies monolith in holomorph
  2. Monolith is characteristic


Proof using given facts

The proof follows from facts (1)-(2).