Difference between revisions of "Monolithic implies linearly primitive"

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(Created page with "==Statement== Suppose <math>G</math> is a finite group. Then, if <math>G</math> is a monolithic group (i.e., it has a ''unique'' minimal normal subgroup) then <math>...")
 
 
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==Statement==
 
==Statement==
  
Suppose <math>G</math> is a [[finite group]]. Then, if <math>G</math> is a [[monolithic group]] (i.e., it has a ''unique'' [[minimal normal subgroup]]) then <math>G</math> is a [[linearly primitive group]], i.e., it admits an [[irreducible linear representation]] over the complex numbers that is also a [[faithful linear representation]], i.e., has trivial [[kernel]]).
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Suppose <math>G</math> is a [[finite group]]. Then, if <math>G</math> is a [[fact about::monolithic group]] (i.e., it has a ''unique'' [[minimal normal subgroup]]) then <math>G</math> is a [[fact about::linearly primitive group]], i.e., it admits an [[irreducible linear representation]] over the complex numbers that is also a [[faithful linear representation]], i.e., has trivial [[kernel]]).
  
 
==Related facts==
 
==Related facts==
  
 
* [[Linearly primitive implies cyclic-center]]
 
* [[Linearly primitive implies cyclic-center]]

Latest revision as of 15:00, 16 July 2011

Statement

Suppose G is a finite group. Then, if G is a Monolithic group (?) (i.e., it has a unique minimal normal subgroup) then G is a Linearly primitive group (?), i.e., it admits an irreducible linear representation over the complex numbers that is also a faithful linear representation, i.e., has trivial kernel).

Related facts