2
$\begingroup$

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!

$\endgroup$
  • $\begingroup$ The phrase you should look up is group presentations. $\endgroup$ – Nate Nov 12 '14 at 23:31
  • 4
    $\begingroup$ 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
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.