1. (a) Prove Fermat's Little Theorem: if p is prime and pa, then a-1 = 1 mod p. (b) Prove that if G is a group of prime order p, then G is cyclic. (c) Prove that if G is cyclic then G is abelian. (d) Prove that a nontrivial group G has no nontrivial proper subgroups if and only if G is finite and of order p where p is prime. (e) Prove that every group of order less than 6 is abelian (you may use facts that we have shown on the worksheets). (f) Give an example of a group of order 6 that is not abelian. 2. Let X be the set C(x], and let G = {-1, 1} with group operation multiplication. Define the group action g . f(x) = f(x) if g = 1 and f(-x) if g = -1. (a) Prove that if g . f(x) = f(x) for all ge G, then f(x) is a polynomial in 12. HINT: PROVE EVERY ODD COEFFICIENT IS 0. (b) Show that the map 9 : C(x] - C(x] by f(x) -> 9 . f(x) is a ring homomorphism for all J EG. (c) Let X be the set C[x], and let G be the group {en |0 9 . f(x) is a ring homomorphism for all J EG. (d) Suppose that f(r) is a polynomial such that g . f(r) = f(x) for all ge G. Prove that f(x) is a polynomial in