Assume and apply times on both sides of your second to last equation and use the minimality of p and q.
Question: Let be cycles of generalized eigenvectors of a linear operator with respect to eigenvalue . Prove that if the initial eigenvectors are distinct, then the cycles are disjoint.
Attempt: Suppose that are (distinct) generalized eigenvectors of and let be the smallest integer such that and .
I tried to argue by contradiction. Say that , then there exist and such that . Then for all and , we have . And then, this needs to imply that are in fact not distinct anymore and thus creates a contradiction ...?