Solve

54. 55. 56. 57. 59. Let G be a group and suppose that H is a subgroup of G with the property that for any a in G we have eff = Ha. (That is, every ele- ment of the form ah where h is some element of H can be written in the form hla for some h: E H.) If a has order 2, prove that the set K = H U aH is a subgroup of G. Generalize to the case that Ial = k. Prove that As is the only subgroup of $5 of order 60. Why does the fact that A 4 has no subgroup of order 6 imply that |Z(A4)| = I? Let G = GL(2, R) and H = SL(2, R). LetA E G and suppose that det A = 2. Prove that AH is the set of all 2 X 2 matrices in G that have determinant 2. Let G be the group of rotations of a plane about a point P in the plane. Thinking of G as a group of permutations of the plane, describe the orbit of a point Q in the plane. (This is the motivation for the name \"orbit.\") Let G be the rotation group of a cube. Label the faces of the cube 1 through 6, and let H be the subgroup of elements of G that carry face I to itself. If 0' is a rotation that carries face 2 to face 1, give a physical description of the coset H0\". The group D4 acts as a group of permutations of the square regions shown below. (The axes of symmetry are drawn for reference pur- poses.) For each square region, locate the points in the orbit of the indicated point under D4. In each case, determine the stabilizer of the indicated point