Help with explation

8:01 f f f Bb . Document (4 - Saved O E . . . 1. Let n be 9 ). Define a function f : Z - G by f (k) = 1 [cos(ke) sin(ke)] - sin(k0) cos(ke) where 0 = 32 (a) Show that f is a homomorphism. (b) Use the first isomorphism theorem to conclude that Z/2nZ is isomorphic to cos(k0) sin(ke)]. - sin(k0) cos(ke) : KEZ). 2. Let n be Determine whether or not D,, x Z/2Z is isomorphic to D2n. @, let n = 24. out let k = 3. Give an example of a nonabelian group G of order n and a normal subgroup # of order k. Prove that H is a normal subgroup of G. Then write down the Cayley table of G/ H. 4. Let r be Let R = {a + bo : a, be Z}. We define addition by (a + bo) + (a' + b'0) = (a + a')+ (6+b')$ and multiplication by (a + bo)(c + do) = ac + rbd + (ad + be + bd)o Show that R is a ring. 5. Let R and S be rings. Define a multiplication on R x S by (r, s) . (r', s') = (rr', ss') and an addition on R x S by (r, s) + (r', s') = (r + r',s + s'). Show that R x S is a ring. BIUDA O