Difference between revisions of "Coprime automorphism group"

From Groupprops
Jump to: navigation, search
(New page: ==Definition== Let <math>G</math> be a finite group. A subgroup <math>H</math> of the automorphism group <math>\operatorname{Aut}(G)</math> is termed a '''coprime automorphism gro...)
 
 
Line 3: Line 3:
 
Let <math>G</math> be a [[finite group]]. A subgroup <math>H</math> of the [[automorphism group]] <math>\operatorname{Aut}(G)</math> is termed a '''coprime automorphism group''' of <math>G</math> if the [[order of a group|order]]s of <math>G</math> and <math>H</math> are relatively prime.
 
Let <math>G</math> be a [[finite group]]. A subgroup <math>H</math> of the [[automorphism group]] <math>\operatorname{Aut}(G)</math> is termed a '''coprime automorphism group''' of <math>G</math> if the [[order of a group|order]]s of <math>G</math> and <math>H</math> are relatively prime.
  
Note that the whole group <math\operatorname{Aut}(G)</math> is very rarely coprime to <math>G</math> in order. {{further|[[Coprime automorphism group implies cyclic with order a cyclicity-forcing number]]}}
+
Note that the whole group <math>\operatorname{Aut}(G)</math> is very rarely coprime to <math>G</math> in order. {{further|[[Coprime automorphism group implies cyclic with order a cyclicity-forcing number]]}}

Latest revision as of 15:57, 22 January 2009

Definition

Let G be a finite group. A subgroup H of the automorphism group \operatorname{Aut}(G) is termed a coprime automorphism group of G if the orders of G and H are relatively prime.

Note that the whole group \operatorname{Aut}(G) is very rarely coprime to G in order. Further information: Coprime automorphism group implies cyclic with order a cyclicity-forcing number