Let Ø:Z50 Z15 be a group homomorphism with Ø(x)=4x. Then, Ker(Ø)= O (0,15,30,45} O None of the choices O (0, 10, 20, 30, 40} O (0, 5, 15, 25, 35, 45}

Suppose now that we have two groups (X,o) and (Y, *). We are familiar with the Cartesian product X x Y of X and Y, but can we also define a binary operation on X x Y such that X x Y is a group? Let's consider two elements (x1, 41) and (x2, Y2) in X x Y. Let denote a function on X x Y.We need to define so that (X x Y,•) is a group. The most natural way to proceed is to define • component-wise: (T1, Yı) • (x2, Y2) = (x1 º Y1, ¤2 * Y2) %3D Now we should show that (X × Y,•) is in fact a group.