Hi, I am in difficulty when I kept trying to do these questions.

Could you give me a help to solve these questions?

Thank you so much.

Q1: Consider the permutation group 37. (a) Can the product of two cycles of length 4 be a cycle of length 6? (b) Can the product of two cycles of length 4 be a cycle of length 7'? Q2: What fraction of the permutations in Sn are cycles of length 71? Q3: Consider two types of triangle: equilateral and isoceles, as depicted: A A C B C B (a) Describe the (rigid) symmetries of an equilateral triangle. (b) Which of the symmetries in part (a) would also be a symmetry of the isoceles tri- angle? Using your answer, justify or refute the following statement: \"Equilateral triangles are more symmetric than isoceles triangles.\" Q4: Consider the dihedral groups. (a) Write the symmetry j(r3j)(r2j)(rj)(r5j) 6 D6 in the form rkj. (b) Is every element of Dn (for any n 2 3) a symmetry of the (circular) disk? Is there a symmetry of the disk that isn't an element of some D\"? Q5: [Writing Question] Which of U12 and U14 are cyclic groups? Justify your answer, providing generators as necessary