Let be a prime. Show that the following matrices are similar in : (the companion matrix of ) and (the Jordan block with eigenvalue 1).
We claim that in fact is the minimal polynomial of . Recall (from this previous exercise) that (the symmetric group on objects) is embedded in by letting a permutation act index-wise on the elements of an ordered basis (i.e. ). Now is in the image of this representation, and in fact is the matrix representing the permutation whose cycle decomposition is . In particular, is not the identity transformation, so the minimal polynomial of is indeed .
So the elementary divisors of are just , and so is similar to the Jordan block .