I Prof. Iarrobino's W54 question 2}: Let H be a subgroup of a group G. The normalizer of H in G is Nc{H]={oEG'|nhn_1 EH1 forahEH}={eEG|an_l=I-I} {a} Prove that NEEH] is a subgroup of G containing H, and prove that H is normal in Ng{H}. [b] Determine the normalizers of the following subgroups: i. The cyclic subgroup K = {T}, where T = [l 2] in 53. ii. H = {e?{12}[34}} in D5. iii. The diagonal subgroup E = {[11,11] I o E 53} in G = S; x 33. iv. In ULgfIEL the subgroup T of diagonal matrices T = { (I; :91) |t E 111\"}