Question

7. Consider the function f on U50 defined by f(x) = [x2] where [2²] repre- sents the equivalence class of Ug that x2 belongs to. Verify that this is a homomorphism and then determine its kernel.

Answer 1

0 We define for each nye Uch is the - set of all positive integers loop less than n and relatively prime to n. uc) is a group of under multiplication : modulo n. How f: Uso - Uso . fra) = [x2] Now we show that f is homomorphism. nie lek x,y & Uso f(-x) = [eya) = [139] = ["][v] = fra). fcy) The kornel is Preu (55) : for=1} = Securso) : (+3=;} Now 4 (50)=(5,5, 7, 9, 5, 15, 17, 19, 21, 23, 27, 29, 51, 33, 37, 39, 41, 43, 47,49). rrow lucso) l = 9 (so) = 50 ($- 5)(--) Now 50 =5x2 = 5x 4 x Š 20 Note: 9(n) is = the number of rositive integers less than rre and prime ton and Q(1)=2 here nye

Now we state another theorem: To a finite grout, the number of elements of orderd is a multiple of to. Q(d) ie we are have to solve 2 = (mad 5o).- Now U(s) is a finite group The From the above theoreem we can say that the number of elements of order 2 is $(2) =1. Now ie z = (med so) trees a unique solution, you can easily check that x=49 (omad so). on the is the solution of O. de kerf - Sw€ 4 (5): [1] =} = [49] = $[47]} Thus Fis homomorphism and menf = $ 481.