Question

1-Assume that f is a homomorphism from (G, ∗) to (H, ⊗). a) If a∈G, why...

1-Assume that f is a homomorphism from (G, ∗) to (H, ⊗).

a) If a∈G, why is f(a^-1) = (f(a))^−1? (To show that w=z^−1 it suffices to show that wz=e.)

b) For all a and b in ker(f) why is a∗b∈ker(f)?

c) Assume that z and w are in the range off. Then there are elements a and b of G such that f(a)=z and f(b)=w. Why is z⊗w in the range of f?

d) Assume that z is in the range of f. This means that z = f(a) for some a ∈ G. Why is z^−1 in the range of f?

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