Additive group of a field implies monolith in holomorph
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.
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
- Abelian implies self-centralizing in holomorph
- Every group is normal fully normalized in its holomorph
- Characteristically simple and NSCFN implies monolith
Proof using given facts
The proof follows from facts (1)-(3).