If A is an n×n matrix over a field, and Ak = I, with k the least positive integer such that this occurs, then must there be some vector v such that { v, Av, A2v, …, Ak−1v } = { Aiv } has k distinct elements in it? In other words:
Must every matrix of finite multiplicative order have a regular orbit?
If A has prime power order, k = pm, then Apm−1−I is nonzero, so its kernel is proper, and everything outside of that kernel is a vector in a regular orbit. Over a finite field of size q, the index of a proper subspace is at least q, so we can even just choose (on average) q random vectors to find one in a regular orbit. Over an infinite field, the same idea roughly says any random vector should work, as long as one can make some sort of "uniformly" distributed choice.
If A has order a product of two prime powers, then I am assured this is true by a (special case) of an exercise in Isaacs's Finite Group Theory. I cannot imagine an argument that does not work for arbitrary orders k, but I also cannot find a convincing proof even for the product of two prime powers. The sum of vectors in regular orbits of the p-parts of A need not themselves be in regular orbits of A. Every matrix (over a finite field) I've tried has a regular orbit.
Assuming this is easy, how does one handle the case where A is an automorphism of a finite group G, and the order of A is a product of two prime powers? In other words:
Prove every automorphism of order paqb of a finite group has a regular orbit.
Assuming the first question's answer is "yes", then what goes wrong for arbitrary orders? Isaacs's book gives an example where the general automorphism can fail to have a regular orbit, but it is impossible to compare this until I have at least some idea of why the two-prime case does work.
A related version of this question is: regular orbits are quite important in permutation and (finite) matrix groups and are a standard technique in several important (solved and unsolved) problems in modular representation theory.
Is there sort of a gentle introduction that puts these techniques in context?
For any individual paper is clear that what they say works, but my picture of this area is incredibly disjointed and I suspect that is not true for everyone. For instance Khukhro has an excellent book on automorphisms of p-groups with few fixed points, and many finite group theory texts have chapters on fixed-point-free automorphisms and the consequences for the group structure of the group being acted upon. However, I haven't found any "textbook" exposition of regular orbits yet.
