4. (a) Let a, b, c, d e C with ad-bc0. Prove that there exists an...

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4. (a) Let a, b, c, d e C with ad-bc0. Prove that there exists an automorphism o of C(z) with o(2) = + (these are called Mobius transformations) cz+d (b) Determine the relationship between composition of Mobius transformations and ma- trix multiplication. (c) Show that the automorphisms o(t)= it and r(t) = t¹ of C(t) generate a group G that is isomorphic to the dihedral group D₁. (d) Let ut+t-4. Show that u is fixed under H. (e) What is [C(t): C(u)]? U= 4. (a) Let a, b, c, d e C with ad-bc0. Prove that there exists an automorphism o of C(z) with o(2) = + (these are called Mobius transformations) cz+d (b) Determine the relationship between composition of Mobius transformations and ma- trix multiplication. (c) Show that the automorphisms o(t)= it and r(t) = t¹ of C(t) generate a group G that is isomorphic to the dihedral group D₁. (d) Let ut+t-4. Show that u is fixed under H. (e) What is [C(t): C(u)]? U=