This is something I am stuck on (it might well be trivial- in which case this is an embarassing question):<br>
Let V be a dimension r vector space over F<sub>p</sub>, the field with p prime elements (I also care about this when V is over Z<sub>n</sub> with n composite). Let t be a <b>given</b> automorphism of V of order q (prime different from p, although I am also interested in more general cases), meaning t<sup>q</sup>=1. <b>edit</b>: It is given that for any non-zero vector v in V, the elements of the set {t<sup>i</sup> v} together generate V, where i is between 1 and q. This implies that 1-t is also an automorphism of V. (thanks David!)
<blockquote><b>Question</b>: What is the order of 1-t? When is it q?</blockquote>
The context is representations of commutator subgroups of knot groups onto vector spaces. Here t is induced by the deck transformations of the infinite cyclic cover.