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- 4. List all the elements of the subgroupin the group under addition, and state its order.The group (U20, X20) = {(1, 3, 7, 9, 11, 13, 17, 19), X20} has subgroups H = {1,3,7,9) and K = {1,11}. qergie laurion Sonnen (a) Explain why both H and K are normal subgroups of U20, and for each of H and K list all of its distinct cosets in U20. (b) For each of H and K, write down the group table of its quotient group in U20, and state a standard group to which the quotient group is isomorphic, briefly justifying your answer. 1010 (The standard groups introduced in M208 are listed on pages 136-137 of the Handbook.) 10% 10 vlad 1990s a alwallal way soiliturThe group (Z4 ⨁ Z12)/<(2, 2)> is isomorphic to one of Z8, Z4 ⨁ Z2, orZ2 ⨁ Z2 ⨁ Z2. Determine which one by elimination.
- Determine all subgroups of R* (nonzero reals under multiplication)of index 2.Are there any other subgroups of = order 3 & 2 ? For his vere'll consider all subsets of 3 and 2 elements of $3 and use equations of fiz and gf² = fg to check if they are subgroups 5-{e, f, f²g, fg, f²g] } All Subsets of 3 elements { e. f₁ f²} = = {e₁ he₁ f₁ g y = Not a Subgp. Since £g¢{ } 2 17 {e, f, f'g]=" { e, f ²₁ gb = {e, f²₁ fg} = 11 {e, 5², 3²g} = 11 11 " "1 11 11 " since f.f²g = fg-=9& { } Since 1²g & {y • Since f²fg. f²f g - f³ g - g 4 2 3 Since f²f²g = f g = fg &{} L> 71 Ee, g, fg} = not a subgp. since gffg)=ggf}=1² #23 Ee, g, fg}= Since g(8²) = g(gf) = f 483 E g, fg, fg}.. Since gf ggf) f €23 So the only subgroups of order. 3 are , 11 2 (1Let G = Z4 × Z6. Compute the factor groups G/((2, 3)) and G/((3, 3)). (In each case, write the result in terms of known finite groups, and explain your answer.)