5. Consider the group S9, a group with 9! = 362, 880 = 27.34.5.7 elements. Find...

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5. Consider the group S9, a group with 9! = 362, 880 = 27.34.5.7 elements. Find a subgroup with 7 elements. (Write down all the elements in the subgroup, and how you thought of your example.) Find a subgroup with 14 elements. (Explain how you thought of your example.) (a) explain (b) (c) Find a subgroup with 120 elements. (Don't list all the elements!... but explain how you thought of your example.) (d) Find the biggest subgroup you can whose order is a power of 3. What is the order of the subgroup you found? (Explain your example clearly, and how you thought of your example.) (e) Explain why there is no subgroup with 25 elements. (f) Do you believe there is a subgroup with 35 elements? We probably haven't learned enough in class for you to answer this definitively, but feel free to share any insights or intuition. 5. Consider the group S9, a group with 9! = 362, 880 = 27.34.5.7 elements. Find a subgroup with 7 elements. (Write down all the elements in the subgroup, and how you thought of your example.) Find a subgroup with 14 elements. (Explain how you thought of your example.) (a) explain (b) (c) Find a subgroup with 120 elements. (Don't list all the elements!... but explain how you thought of your example.) (d) Find the biggest subgroup you can whose order is a power of 3. What is the order of the subgroup you found? (Explain your example clearly, and how you thought of your example.) (e) Explain why there is no subgroup with 25 elements. (f) Do you believe there is a subgroup with 35 elements? We probably haven't learned enough in class for you to answer this definitively, but feel free to share any insights or intuition.

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a One example of a subgroup with 7 elements in S9 is the cyclic subgroup of S9 generated by an eleme... View the full answer