U(n) is defined to be the group with the numbers relative prime up to n (such that the gcd between n and an x in U(n) is 1). For example U(9)=1,2,4,5,7,8. Uk(n) is defined as the set of x's that are an element of U(n) s.t. x mod k=1. For example U7(105)={1,8,22,29,43,64,71,92}.
Let k|n. Consider ƒ:U(n) to U(k) defined by ƒ(x)=x mod k. What is the relationship between this homomorphism and the subgroup Uk(n) of U(n).
Sorry there should have been a k in the second u and not an n. So the kernel is the set of elements x in the domain that map to the identity element in the co domain. Since Uk(n) = x element of U(n) s.t. x mod k = 1, then Uk(n) would just be the kernel of f. Is this what you are saying because this seems right, but really easy for some reason.