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 F_p, the field with p prime elements (I also care about this when V is over Z_n 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^q=1. It is given that:

1. 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.
2. 1-t is also an automorphism of V.

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.

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 F_p, the field with p prime elements (I also care about this when V is over Z_n 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 t^q=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.