Question

Town Hall with Pe CheapOair Cross X Amouncements x Crowdmark X proiecte X + - ssments/homework-7-4c94f Task 32 (10 points) a) Find the homomorphism f: Z → Z defined by setting /(1) = 1. (You do not need to verty that it's a homomorphism.) That is complete the following (remember, it has to be a homomorphism, so it must satisfy the homomorphism property): 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Also find K = Ker(), and list all cosets of K. What do you notice about the relationship of the function images of elements that come from the same coset? To what familiar group is 215/K isomorphic? b) Now find a second homomorphism g: Z1 + Zs by setting g(1) = 3, so complete 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 go 3 Also find K = Ker(9) and list all cosets of K. What do you notice about the relationship of the function images of elements that come from the same coset? To what familiar group is 215/K isomorphic? + Drag and drop your mages or click to browser

Answer 1

* Since under a homomorphism, feny)= f(u) fly). and in zun, k= 1.K & KERA = f(k)=kfci) kezhn.

(1=1 o fcko= k fci)-k.lk noco 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 101 2 3 4 5 6 7 8 9 10 11 12 13 14 » K= konf = 203 So the caset ark=faroy-a os K={03, it k={13, 2+K= {23, 3+K= {3} 13+K= {133, 14+K={143 This we see all the Cassts have same number of elements as much Kernel have also flatk) = f(b)+ 1.e. all the elements in a caset have same 2415 = 24 x 2415 image 203 . (

f01)-3 - fck) = K. f(0) = 3K f(K) = 3K tke mis 10 lo 1 3 2 6 3 9 4 12 5 0 6 3 7 6 § 9 9 12 10 o 3 12 6 27771fiilit111LLLL 2 K= kerf = {0,5,10} now at ku fato, at 5, arvok = {2, at 5, atio so K= 10,5,103 i+K={ 1,6,li} 2+k= {2, 7, 12} 3+K={3, 8, 134 utk = {4, 9, 143 These are the only casets as Indexaf Kin2lig is 5. now we can see that all the elementsin se a Caset have same image. 2+K= {2,7,123 —>{o} as for example - - es b - as quatient graup of Cyclic graups are Cyclic .