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 an 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. It is given that:
<ol><li>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.</li>
<li> 1-t is also an automorphism of V.
</ol>
<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.