3. For any a eE Z, let als denote (a] in Z6 and let (a)2 denote (a] in Z2. Given that the mapping 0: Z6 Z2 defined by 0(lale) = [a]2 is a homomorphism. %3D (a) Find ker 0. (b) Find Zg/ker0. (c) Find an isomorphism from Z6/kere to Z2.

Exercise 4.2.5. Let q: Z6 → Z2 × Z3 be the map q([k]6) = ([k]2, [k]3). (1) Show p is well-defined. (2) Show p is a homomorphism. (3) Show q is injective. Why is this enough to conclude q is an isomorphism?

{ (80) : a, b,ce z}. and for all such A E L let f, g, h: L→ Z be given by b f(A) = a, g(A) = tr(A), h(A) = det(A) (a) Show that f is a surjective homomorphism. (b) Show that g is additive, but not multiplicative. (c) Show that h is multiplicative, but not additive. 7. Let L =

Please avoid using theorems that are uneccesarily advanced