0
$\begingroup$

I found on the Internet (here if you speak French) without any proof nor reference the following result.

Let $G$ be a non abelian simple group. Then $G$ is generated by a pair of elements.

In other words, there exists $(g_1,g_2)\in G^2$ such that $G=\langle g_1,g_2\rangle$.

How someone would prove such a result? Do you have any references where this is done? Is it a known fact (I have found nothing on the Internet)?

  • 0
    As far as I recall, this is one of those "simple" facts that are only known by virtue of the classification. – 2017-01-18
  • 0
    @TobiasKildetoft it would be a fascinating claim to have a more direct proof for, though. I wonder if the number of generators can be proved to be below some other limit without using the classification of all groups. – 2017-01-18
  • 0
    [here](http://mathoverflow.net/questions/59213/generating-finite-simple-groups-with-2-elements/59300#59300) is a link containing some discussion of the point and several references. – 2017-01-18
  • 0
    @lulu Thank you for you link, a lot of informations there ! – 2017-01-18
  • 2
    It is proved in a 1962 [RobertSteinberg's article](http://cms.math.ca/10.4153/CJM-1962-018-0) in the *Canadian Journal of Mathematics*. – 2017-01-18
  • 1
    Googling your title gives a lot of literature, so i wonder why you said "I have found nothing on the Internet". – 2017-01-18
  • 1
    @DietrichBurde Because I didn't Googled it in English... (my bad) – 2017-01-18
  • 0
    @Bernard Steinberg's article only covers the finite simple groups that were known in 1962. – 2017-01-18
  • 0
    Really? I didn't read it, just looked at the presentation of the results. Maybe I read too hastlily. – 2017-01-18

0 Answers 0