# Difference between revisions of "Coprime automorphism group"

From Groupprops

(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 be a finite group. A subgroup of the automorphism group is termed a **coprime automorphism group** of if the orders of and are relatively prime.

Note that the whole group is very rarely coprime to in order. `Further information: Coprime automorphism group implies cyclic with order a cyclicity-forcing number`