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- Find number of elements of order 2 in A5 (Alternating group)Q6/ Give example of a group such that has element of order 7 and nurmal subgruup of Order 3.List all groups of order up to 12, Then *Note that the groups of order up to 12 must contain cyclic group, symmetric group, alternating group and dihedral group, additional groups are also preferred
- The multiplicative group of positive rational number is cyclic If its true give detailed proof if its not ture give exampleDetermine all subgroups of R* (nonzero reals under multiplication)of index 2.Check whether the following are groups. Justify your answer. (i) (G, *) where G= {1, 2, 4, 8}, and * denotes multiplication modulo 12. That is, a*b=Remainder obtained when ab is divided by 12. (ii) (G, A) where G = {6,12,18,24}, and A denotes multiplication modulo 30. 3.
- (5.2) Challenge Modern Applied Algebra. Show work.let G={e, a, a^2, a^3, a^4, a^5, a^6, b, ab, a^2b, a^3b, a^4b, a^5b, a^6b}. related by the equation ba=a^2b, fined order of each element and proper nontrivial subgroups of G.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 (1