The symmetric group on 7 symbols, S7, has 7!=5040 elements. Consider the following questions without employing Sage, based on what we know about orders of elements of permutation groups:
• What is the maximum possible order?
• How many elements are there of order 10?
• How many elements are there of order 1?
• How many elements are there of order 2?
• What is the smallest positive integer for which there is no element with that order?
Give some serious thought to each question before firing up Sage.
Now, compute how many elements there are of each order using the .order() method, and then embed this into a list comprehension which creates a single list of these counts. You can check your work (or check Sage) by wrapping this list in sum() and hopefully getting 5040.
Comment on the process of studying these questions first without any computational aid, and then again with Sage. For which values of n do you think Sage would be too slow and your mind quicker?