Problem 3.[10 points.] Let D. be the dihedral group of order 2n. Let H = (r)...
Let D2n be the dihedral group of order 2n i.e. the group of sysmmetries of the regular n-gon. Let H be the set of rotations of the regular n-gon. Prove that H D2n.
2. Let n 2 3, and G D2n e,r,r2,... ,r"-1,s, sr, sr2,..., sr-'), the dihedral group with 2n ele- 3, ST, ST,..,ST ments. We let R-(r) denote the subgroup consisting of all rotations. (a) Show that, if M is a subgroup of R, and is in GR, then the union M UrM is a subgroup of G. Here xM-{rm with m in M) (b) Now take n- 12 and M (). How many distinct subgroups does the construction in (a)...
10. Let G = D. be the dihedral group on the octagon and let N = (r) be the subgroup of G generated by r4. (a) Prove that N is a normal subgroup of G. (b) If G =D3/N, find G. (c) Using the bar notation for cosets, show that G = {e, F, 2, 3, 5, 87, 82, 83}. Hint: Show that the RHS consists of distinct elements and then use part (b). (d) Prove that G-D4. Hint: It...
abstract algebra show your work 3. Let H be a subgroup of G with |G|/\H = 2. Prove that H is normal in G. Hint: Let G. If Hthen explain why xH is the set of all elements in G not in H. Is the same true for H.C? Remark: The above problem shows that A, is a normal subgroup of the symmetric group S, since S/A, 1 = 2. It also shows that the subgroup Rot of all rotations...
Problem 3. Subgroups of quotient groups. Let G be a group and let H<G be a normal subgroup. Let K be a subgroup of G that contains H. (1) Show that there is a well-defined injective homomorphism i: K/ H G /H given by i(kH) = kH. By abuse of notation, we regard K/H as being the subgroup Imi < G/H consisting of all cosets of the form KH with k EK. (2) Show that every subgroup of G/H is...
Let D4 be dihedral group order 8. So D4={e, a, a^2, a^3, b, ab, a^2b, a^3b}, a^4 = e, b^2= e, ab=ba^3; A. FIND ALL THE COSETS OF THE SUBGROUP H= , list their elements. B. What is the index [D4 : H] C. DETERMINE IF H IS NORMAL
Let De be the dihedral group of order 12. In other words, Do = {e,r,r2, m3, 74, 75, s, rs, rs, rºs, r*s, r® s} where p6 = 92 = e and sr = r-1s. a. Is H = {1, s, sr, sr2} a subgroup of Do? Why or why not? b. Is K = {1, 8, 73, r3s} a subgroup of Do? Why or why not?
I help help with 34-40 33. I H is a subgroup of G and g G, prove that gHg-1 is a subgroup of G. Also, prove that the intersection of gH for all g is a normal subgroup of G. 34. Prove that 123)(min-1n-)1) 35. Prove that (12) and (123 m) generate S 36. Prove Cayley's theorem, which is the followving: Any finite group is isomorphic to a subgroup of some S 37. Let Dn be the dihedral group of...
Let G be a group of order 231 = 3 · 7 · 11. Let H, K and N denote sylow 3,7 and 11-subgroups of G, respectively. a) Prove that K, N are both proper subsets of G. b) Prove that G = HKN. c) Prove that N ≤ Z(G). (you may find below problem useful). a): <|/ is a normal subgroup, i.e. K,N are normal subgroups of G (below problem): Let G be a group, with H ≤ G...
Problem 3 () (2 marka) Prove that the group R and the circle group St are not isomsorphic to each other. Hind เตบ๐s fad element of order 2 m S., Hou about RV (a)(2marks) Let n 2 be an integer, give an escample (including explanatlon) of a group G and a subgroup FH with IG: H-nsuch that H is not normal in G. (iii) (S marks) Let G-16:l : a,b,c ER, a 7.0, eyh 아 You are given that G...