This is something I am stuck on (it might well be trivial- in which case this is an embarassing question):  

Let V be a dimension r vector space over Fp, the field with p prime elements (I also care about this when V is over Zn with n composite). Let t be a **given** automorphism of V of order q (prime different from p, although I am also interested in more general cases), meaning tq=1. **edit**: It is given that for any non-zero vector v in V, the elements of the set {ti v} together generate V, where i is between 1 and q. This implies that 1-t is also an automorphism of V. (thanks David!)



> **Question**: What is the order of 1-t? When is it q?


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.



Order of "one minus automorphism"
Perhaps the answer is just that the order is what it is, and that sometimes it's q. Do you have a compelling reason to believe that there is any more structure to your question than that?


Here's an example of the situation. $V$ could be the finite field $F$ with $p^r$ elements, and $q$ could be a prime divisor of $p^r-1$. Now $F^\times$ is cyclic of order $p^r-1$ so will contain elements of order $q$. Let $t$ be such an element. Multiplication by $t$ gives an endomorphism of $V$. The minimal polynomial of $t$ over the prime subfield will have degree dividing $r$, and if $q$ doesn't divide $p^s-1$ for any $s$ less than $r$ then it will have degree equal to $r$. So if $q\geq r$ and $q$ divides $p^r-1$ but not $p^s-1$ for any $s$ less than $r$, and such primes are easy to find (for example take $r$ to be 1---or let $r$ be prime; then for any given $p$ there is almost always a $q$) then the conditions of your question are satisfied (this "degree $r$" business is just making sure the vectors span). Now you're asking what the order of $1-t$ is, and it will be some divisor of $p^r-1$ and it might be $q$ and it might not be. 


For example let's choose everyone's favourite prime $p=691$, let $r-1$, let $q=23$ be the biggest divisor of $p-1$, and let's consider the 22 primitive 23rd roots of unity in $\mathbf{Z}/691\mathbf{Z}$. For each of these $t$, let's ask if 1-t is also a 23rd root of 1. The answer is a resounding "sometimes" (namely when $t=20$ or 672).


