Let f : G → H be a homomorphism of groups. Let

ker(f): = {g ∈ G|f(g) = e}

where e ∈ H is the identity of H.

(a) Show that ker(f) is a subgroup of G. (ker(f) is called the kernel of f).

(b) Show that ker(f) is a normal subgroup.

(c) Show that f is injective if and only if kerf(f) = {e}