Subgroups of SL_2 over a finite field

Dear Readers:

Let K be a field. Let G = SL_2(K). Let H be a proper subgroup of G.
Then, unless I am very mistaken, the index of H in G is at least |K|
+1, where |K| is the number of elements of the field K.

(Oh, assume |K|>11 - there are some exceptions for |K|<=11.)

(a) For K finite, this is supposed to be a theorem of Galois's - or at
least Dickson (1899 or so) says so. Is this right? I've heard it being
credited to Frobenius - what is the reason for this mistake (if it is
a mistake)? Does the statement follow easily from Frobenius's theorem
about the dimension of the smallest-dimensional representation of
SL_2(K)?

(b) For K infinite, the statement says that there are no subgroups of
finite index. Is there a simple proof of this? (Do you know of a
complicated proof, for that matter?)

Harald
h.andres.helfgott@xxxxxxxxxxxxx

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