1. The problem statement, all variables and given/known data Let f : G → H be a homomorphism of Abelian groups. 1. Show that f (0) = 0. 2. Show that f (−x) = −f (x) for each x ∈ G. 2. Relevant equations 3. The attempt at a solution My background in topology / group theory is next to nothing. 1. Show that f(0) = 0. My attempt is as follows: f(x+y) = f(x) + f(y) Let y = 0 f(x+0) = f(x) + f(0) f(x) -f(x) = f(x) -f(x) + f(0) 0 = 0 + f(0) 0 = f(0) Is this a valid proof? 2. I tried the following: f(-x) = f(-1 * x) = f(-1)*f(x) = -1 * f(x) = -f(x) I'm fairly certain that for a homomorphism, f(1) = 1. Is that correct? I'm guessing that f(-1) = -1, though I'm not sure whether this is correct. Can somebody please provide me with some comments or suggestions? Thank you!