Although this is intentionally a rather informal first introduction to the world of permutations, I was too sloppy at some places - big thanks to "tobit4517" for pointing that out. In proposition 10, at 12:18, one has to exclude compositions of 1-cycles. Otherwise, one could, e.g., add the identity id = (1) (2) ... (n) any arbitrary number of times, thereby destroying uniqueness of the cycle decomposition. Then, proposition 10 does not apply to the identity. Also the term "uniqueness" is not clearly defined. A more formal proof of the following fact could and should have been presented: a cycle that starts with, e.g., 1 must reach 1 again at some point, thereby completing the cycle. The terms "length of a cycle" and "disjointness of cycles" were not properly defined.
Although this is intentionally a rather informal first introduction to the world of permutations, I was too sloppy at some places - big thanks to "tobit4517" for pointing that out.
In proposition 10, at 12:18, one has to exclude compositions of 1-cycles. Otherwise, one could, e.g., add the identity id = (1) (2) ... (n) any arbitrary number of times, thereby destroying uniqueness of the cycle decomposition. Then, proposition 10 does not apply to the identity.
Also the term "uniqueness" is not clearly defined.
A more formal proof of the following fact could and should have been presented: a cycle that starts with, e.g., 1 must reach 1 again at some point, thereby completing the cycle.
The terms "length of a cycle" and "disjointness of cycles" were not properly defined.