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Exercise 1.3.19 (Order of the elements of $S_7$)
Find all numbers such that contains an element of order . [Use Exercise 15.]
Answers
Proof. Let be any element of .
As in Exercise 19, the ways to decompose in some of integers are
Each way corresponds to a type of cycle decomposition, so the cycle decomposition of is one of the following forms
| type | ||
| 1 | ||
| 2 | ||
| 2 | ||
| 2 | ||
| 3 | ||
| 6 | ||
| 6 | ||
| 3 | ||
| 4 | ||
| 4 | ||
| 12 | ||
| 5 | ||
| 10 | ||
| 6 | ||
| 7 | ||
So the orders of the permutations of are . □
Verification with Sagemath:
sage: G = SymmetricGroup(7); G
Symmetric group of order 7! as a permutation group
sage: s = set()
sage: for sigma in G:
....: s.add(sigma.cycle_type())
....: s
....:
{[1, 1, 1, 1, 1, 1, 1],
[2, 1, 1, 1, 1, 1],
[2, 2, 1, 1, 1],
[2, 2, 2, 1],
[3, 1, 1, 1, 1],
[3, 2, 1, 1],
[3, 2, 2],
[3, 3, 1],
[4, 1, 1, 1],
[4, 2, 1],
[4, 3],
[5, 1, 1],
[5, 2],
[6, 1],
[7]}
sage: s = set()
sage: for sigma in G:
....: s.add(lcm(sigma.cycle_type()))
....: s
....:
{1, 2, 3, 4, 5, 6, 7, 10, 12}
