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a) (2.5 pts) Show that the set C* = E C | z # equipped with the usual complex multiplication ". is a group. b) (2.5 pts) Show that the set S¹ = E C | z = 1} equipped with the usual complex multiplication 6.' is a subgroup of C*. c) (2.5 pts) Show that for every n E N, the group S¹ has a subgroup isomorphic to Zn. d) (2.5 pts) Show that the group S¹ has a subgroup isomorphic to Z. 2. (10 pts) Solve the following problems. a) (3 pts) Show that Dn contains a subgroup isomorphic to Zn and for every d that divides n the group Dn contains a cyclic subgroup of order d. b) (4 pts) Show that Dn contains a subgroup isomorphic to Dk iff k divides n. c) (3 pts) Find all subgroups of Dn that are isomorphic to 22 (Hint: Consider two cases: (i) n - odd, and (ii) n-even.) 3. (15 pts) Let GL (2,R) = {M E Mat (2,R) | det (M) # 0} and let O (2) = {M € GL (2,R) | MMT=1] = a) Show that O (2) is a subgroup of GL (2,R) b) Show that O (2) = - 0 € A € - cos 0 0 1 E GL 0 € 0,27) c) Let SO (2)= {ME0(2) = | det (M) 1} Show that SO (2) is a subgroup of O (2) and SO (2) = { sin cos 0 0 - cos sin 0 0 € GL 0 E . d) Show that SO (2) is isomorphic to S¹ and using Problem #1 conclude that O (2) has subgroups isomorphic to Zn and TL. e) Prove that the map 4 : Dn O (2) defined on generators = sin A - COS sin 0 0 and = extends to an injective homomorphism of Dn into O (2). Conclude that O (2) contains a subgroup isomorphic to Dn for all n E N. 4. (15 pts) Let GL (2,C) = {M € Mat (2,C) | det (M) # 0} denotes the general linear group of com- plex matrices of size 2 X 2. a) Show that H = {1" 21 22 E Mat (2,C) | 21 2 + 0 is subgroup of GL (2,C). a b) Let HI = {a+bi+cj+dk | a, b, c, d E R} denotes a ring of quaternions. Show that H* = {q E HI | 9* 0} equipped with the usual quaternionic multiplication ". is a group. c) Let u : H*-H be a map given by u + bi + + = 21 22 where 21 = a + bi, 22 = c+ di and Z denotes complex conjugate of Z. Show that u : H*-H is an isomorphism of groups. d) For q E H, define = dk. Show that for p,qEH, Hint: Notice that = det q + bi + cj dk and 21 a+bi,z2=c+di. = a + = = e) Show that the set S³ = = equipped with the usual quaternionic multiplication ". is a subgroup of H*.

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