1. Show that the function from G to G given by a > a 1 is an isomorphism iff G is abelian. 2. Determine if the following statements are TRUE or FALSE. Give reasons for your answers in each case. a) If H is a subgroup of G and a, b E G are such that a * b, then (a * H) n (b * H) = 0. b) If H is a subgroup of G and a E G is such that a * H has 4 elements, then H has 4 elements. c) If H is a subgroup of a finite group G, then for any a E G, the left coset a * H has the same number of elements as the right coset H * a. d) All groups of order 7 are isomorphic to (77, +). e) All groups of order 6 are isomorphic to (Z6, +). f) The group S4 does not have an element of order 13. g) The group S4 does not have an element of order 24. h) (*) The group S4 does not have an element of order 12. 3. a) Verify that is a group with matrix multiplication. b) Show that x E R and > > 0 is a subgroup of G. c) An element O of G can be represented by the point (r, y) in the right half-plane {(x, y) | x, y E R and x > 0}. Draw the partitions of the right half-plane into left and into right cosets of H. 4. Let V be a vector space. Prove, directly from the axioms, that if a E R and v E V are such that a . v = o, then either a = 0 or v = 0. Hint: If a # 0, how can you manipulate the equation a . v = o?5. Consider the set Ro of all sequences with addition defined as follows: if (an)nEN, (bn)nEN ER , then (an)nEN + (bn)nEN = (an + bn)nEN, (1) and scalar multiplication defined as follows: if a E R and (an)nEN E Ro, then a . (an)nEN = (aan)nEN. (2) Prove that Ro is a vector space with the above addition and scalar multipli- cation. 6. Let X and Y be two subspaces of a vector space V. Define the set X + Y to be X+Y = {cty lacX, yEy}. Show that X + Y is a subspace of V