If $A$ is an $n\times n$ matrix over a field, and $A^{k} = I$, with $k$ the least positive integer such that this occurs, then must there be some vector $v$ such that $\{v,Av,A^{2}v,\dots,A^{k-1}v\}$ has $k$ distinct elements in it? In other words:



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> Must every matrix of finite multiplicative order have a regular orbit?
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If A has prime power order, $k = p^{m}$, then $A^{p^{m-1}}-I$ 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.




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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:



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> Prove every automorphism of order $p^{a}q^{b}$ of a finite group has a regular orbit.
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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.




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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.



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> Is there sort of a gentle introduction that puts these techniques in context?
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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.



Linear algebra and regular orbits
For your first question, I presume you also wish to insist that $k$
be the *least* integer such that $A^k=I$. The matrix $A$ is then similar
over your field to a direct sum $B\_1,\ldots,B\_m$ of companion matrices
of (over your field $F$) factors of $X^k-1$, say $f\_1,\ldots,f\_m$.
Then $F^n$ decomposes as a direct sum of subspaces where $A$ acts
cyclically with generator $v\_i$ annihilated by $f\_i(A)$. 


Let $v=v\_1+\cdots +v\_m$. Then
for a polynomial $g$, $g(A)v=0$ if and only if $g(A)v\_i=0$ for all $i$
if and only if $f\_i\mid g$ for all $i$.
Hence $F\mid g$ where $F=f\_1\cdots f\_m$. But then $F(A)u=0$ for all $u$
(so that $F$ is the minimum polynomial of $A$).
If $A^l=I$ where $l < k$ then $F\mid(X^l-1)$ and then $A^l-I=0$,
contrary to hypothesis. So yes, $A$ has a regular orbit.


