Please solve it as fast as possible. Abstract Algebra

12. (30 PTS) PROOFS: Do three of the four following proofs. Be sure to indicate which ones you are doing! (Each proof is worth 10 points.) A. Let G be a group of order 99. Prove that, for any element 9 E G , there must be an element x in G for which :52 = g. B. Suppose that H = {9, a} is a normal subgroup of G. Show that ab = bat for all .5 in G. C. Let R be a commutative ring with identity, and suppose that a E R. Show that the element (1 cannot be both a unit and a zero divisor. D. Let F be a eld. Prove that the set F ' = {x E F: x at 0} forms an Abelian group under multiplication (whatever the multiplication is in F)