# Additive group of a field implies monolith in holomorph

From Groupprops

## Contents

## 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

### 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)

- 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

- 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).