As we know,if G is simple and ︱G︱=︱PSL(2,q)︱,then G is isomorphic to PSL(2,q).My question is if there is a character free proof for that.If there is one,how to do it?
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There is a proof when q is a prime, due to Frobenius. I give a somewhat simplified version of it in a note that will appear in the American Mathematical monthly this October. (See my question 61348). The general problem seems, as has been noted, far less tractable. 


See Borcherds' answer to a related question: He mentions that "It is usually extraordinarily difficult to prove uniqueness of a simple group given its order, or even given its order and complete character table." I expect the answer to your question is no, there is no simple proof and in particular, no proof that is genuinely characterfree. 

