^{A.1 solve problem and show work.}

A. Examples of Homomorphisms of Finite Groups 1 Consider the function f : zg - Z4 given by 0 1 2 3 4 5 6 7 f=01230123 Verify that f is a homomorphism, find its kernel K, and list the cosets of K. [REMARK: To verify that f is a homomorphism, you must show that f(a + b) = f(a) + f(b) for all choices of a and b in zo; there are 64 choices. This may be accomplished by checking that f transforms the table of zg to the table of z4, as on page 136.]