(7) Let I be a line through 0 in R and a > 0 be any angle. Let Fi be the reflection nin I and Ra be the rotation by a around 0. Let H = (Ra, Fi) C O(2) be the subgroup of O(2) generated by Ra and Fi. Prove that H = {FFR. | k = 0, 1; n EZ}. Hint: Show that Fio Ra = R_a . Fi.27: Want to show = SF" RJR: 1=oils 11 H K Step Qi FERICRICH, CREEFORWEEKS H BS a Subgo & F, Rx EH = ) F" RR EH JK CH. Step 2: Show K B a subgroup First show hillt Rx F = F RX . . . Next subgroup test for k Va,bek, a = F" Rx ki , b = F 1 2 R, k2 nipki- kene aby = FRd 4 if 12= 0, this is F"IR, KIkiEK if n2= 1, this is F1" R. R2 - K , ( since Rx F = FRL - m L) either nithz = 2 Fuithe = Ro or nith 2 = 1, this is FRx , ke - ky - Jab"'EK, K is a subgp Step (3): Since F, RIEK & H is the smallest subge containing F, RL - HSK. - H = K