I know that SL(2,Z) is generated by S and T, where

S= (0 -1

1 0)

and T= (1 1

0 1).

But how can I show that everyone element of G (the group generated by S and T) is in SL(2,Z)?

Also, let FcH (upper half-plane) be defined as F= {z in C: abs(z)>1, abs(Re(z)<1/2)}.

How can I draw a picture of F? Which linear fractional transformations correspond to S and T (as given above)?