Prove if it is a group or not. 1. G = {x ≤R | 0 < x < 1},x * y = xy 1-x-y+2xy

If f: Z -> Z is the map defined by f(x) = 2x. Is f a group homomorphism when the group operation on Z is addition? How would you prove it?

8 1 C A B (ii), (iii) (i), (ii) (i) (iv) (i), (iv) Suppose f: A B is a non-zero group homomorphism. Which of the following are true? (i): If A = Z36, B = Z6, f(2)=4, then f(10) = 2. (ii): If A = B = Z20, f (7) = 9, then f(1) = 6. (iii): If f is an isomorphism, then ker(f)# {e}. (iv): If A = U10, B = Z₁, f(3) = 2, then ker(f) = {1}.

|Let G and H be groups, and let 0 : G > H be a homomorphism. The set {x E G| 0(x) |e} is called the kernel of 0 and is denoted by ker (0). Show that ker (0) is a subgroup |of G -