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

From Groupprops
Jump to: navigation, search
(Created page with '==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 gr…')
 
(Related facts)
 
(One intermediate revision by the same user not shown)
Line 1: Line 1:
 
==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]]
* [[Characteristically simple implies CSCFN-realizable]]
+
* [[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

Statement

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

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

Related facts

Analogues for other kinds of groups

Other related facts

Facts used

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

Proof

Proof using given facts

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