# Difference between revisions of "Additive group of a field implies monolith in holomorph"

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==Statement== | ==Statement== | ||

− | 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. | + | 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 [[fact about::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]]. | Then, the [[fact about::holomorph of a group|holomorph]] of <math>G</math> is a [[monolithic group]] with <math>G</math> its [[monolith]]. | ||

==Related facts== | ==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) | ||

+ | |||

+ | ===Other related facts=== | ||

* [[Characteristically simple and non-abelian implies automorphism group is complete]] | * [[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== | ==Facts used== |

## Latest revision as of 22:04, 2 September 2009

## 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).