It is well known that any group is a homomorphic image of a free group. I want to know more about this theorem when $G$ is a finite simple group. Does there exist any reference to state about it? Thanks in advance!

  • $\begingroup$ The phrase you should look up is group presentations. $\endgroup$ – Nate Nov 12 '14 at 23:31
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    $\begingroup$ I think the question is too broad or too vague. You need to say more precisely what you would like to know. There is a lot of literature on presentations of the finite simple groups. $\endgroup$ – Derek Holt Nov 13 '14 at 1:07
  • $\begingroup$ The kernel is maximal of finite index?... I wonder what happens when you intersect all possible kernels for all such groups...hmm... $\endgroup$ – user1729 Nov 13 '14 at 8:05
  • $\begingroup$ I think this is a fact that "any finite simple group is generated by two elements, which one of them is an involution". I think there is a relationship between this fact and this question. Am I right? $\endgroup$ – sebastian Nov 13 '14 at 8:57
  • $\begingroup$ Dear user1729, is it possible to explain your mean more? thanks! $\endgroup$ – sebastian Nov 17 '14 at 8:00

Based on the comment you added to your question, one relationship is that any group that is generated by $k$ elements is a homomorphic image of a free group of rank $k$. This is an immediate consequence of the universal property for free groups.


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