I need help on those 5 response question in bullet point in the last screenshot out of 7 below please!

1 Cosets and Lagrange 1. Let G be a group of order 60. What are the possible orders for the subgroups of G? Explain your reason. 2. Consider 5'3, the group of permutations on the set {1, 2, 3} and let H = {(1), (1 2)}. (3.) Determine the right eosets of H. (b) Determine the left cosets of H. 3. Let 11,!) E G and H, K be subgroups of G. If eff = bK, prove that H = K. 4. Suppose G is a. nite group of order n, and m is an integer relatively prime to :1. If g E G and 9'\" = 89, prove that g = 69. 2 Normal Subgroups and quotient groups 5. Let G1, G2, be two groups and lets up : G1 > 02 be a map such that 90(91 *al 92) = M91) *02 99(92)- Dene the set {9 6 6'1 now) = 602} Prove the given set is a normal subgroup of G1. 6. Suppose H and K are both normal subgroups of G and H F] K = ea. Prove Me = kh for all h E H and k E K. 3. A response to your choice of 5 of the questions below (each response should be thoughtful, single spaced, size 12pt font) What mathematical ideas explored in this class connect or relate to K12 curriculum. Further explore at least one connection you notice, explain it fully, and explain how it enlightened your way of thinking of mathematics. Choose one interesting problem [proposition /theorem from the text of medium difculty that was not done. Describe why you nd it interesting. Then either solve it, or nd a solution online and work through it, using your own understanding to critique that solution and improve it. How has your mathematical imagination been enhanced as a result of taking this class? Give at least three examples. Reect on your overall learning in this class by describing an interesting idea that you learned, why it was interesting. What does it tell you about doing or creating mathematics? Take one practice/ deeper dive problem you have worked on this semester that you struggled to understand and solve. Explain the idea of the problem and explain how the struggle itself was valuable. Consider one mathematical idea from the course that you have found particularly interesting, and explain why it is interesting to you. Your answer should explain the idea in a way that could be understood by a classmate who has taken classes X and Y but has not yet taken this class. How did the ideas of this course enlarge your sense of what it means to do mathematics