# finite simple groups and free groups

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!

• The phrase you should look up is group presentations. – Nate Nov 12 '14 at 23:31
• 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. – Derek Holt Nov 13 '14 at 1:07
• The kernel is maximal of finite index?... I wonder what happens when you intersect all possible kernels for all such groups...hmm... – user1729 Nov 13 '14 at 8:05
• 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? – sebastian Nov 13 '14 at 8:57
• Dear user1729, is it possible to explain your mean more? thanks! – 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.