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Suppose <math>G</math> is isomorphic to the [[fact about::additive group of a field]]. Equivalently, <math>G</math> is a [[fact about::characteristically simple group]] that is also an [[abelian group]]. In particular, <math>G</math> is either an [[elementary abelian group]] or a direct sum of copies of the rationals.
Then, the [[fact about::holomorph of a group|holomorph]] of <math>G</math> is a [[monolithic group]] with <math>G</math> its [[monolith]].