# Additive group of a field implies monolith in holomorph

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## Statement

Suppose is isomorphic to the Additive group of a field (?). Equivalently, is a Characteristically simple group (?) that is also an abelian group. In particular, is either an elementary abelian group or a direct sum of copies of the rationals.

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

## Related facts

- Characteristically simple and non-abelian implies automorphism group is complete
- Characteristically simple implies CSCFN-realizable

## Facts used

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

## Proof

### Proof using given facts

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