MHB Solving for Distinct Vectors of G: $\mathbb{F}_p$, n, and v

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The discussion focuses on determining pairs of p and n for which a fixed vector v and a matrix M can be defined such that the k-fold composition of the function G produces distinct vectors in $\mathbb{F}_p^{n}$. The function G is defined as G(x) = v + Mx, and the goal is to ensure that the outputs G^k(0) for k ranging from 1 to p^n are all unique. The problem requires a deep understanding of linear transformations and modular arithmetic. Participants are encouraged to share insights and strategies for approaching the problem, emphasizing the importance of considering the properties of the matrix M and the vector v. The thread highlights the complexity of achieving distinct outputs in the context of finite fields.
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Let $\mathbb{F}_p$ denote the integers mod p and let n be a positive integer.
Let v be a fixed vector $\in \mathbb{F}_p^{n}$, Let M be a nxn matrix with entries from $\mathbb{F}_p$. Define G:$\mathbb{F}_p^{n} \to \mathbb{F}_p^{n}$ by
G(x) = v + Mx. Define the k-fold composition of G by itself by $G^{1}(x) = G(x)$
and $G^{k+1} = G (G^{k}(x))$ Determine all pairs p,n for which there exists a vector v and a matrix M such that the $p^n$ vectors of $G^{k}(0), k=1,...,p^{n}$ are distinct.
 
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I've solved it but it will take a bit to type the justification for some of the points but contained in the spoiler are general tips for how to think about the problem
This is a pretty interesting question.
Basically it is asking, for which matrix group $GL_{n+1}(F_p)$, can you generate all the elements in $(F_p)^n$ by iterating a matrix (raising it to the power), and get all the elements in $(F_p)^n$.

Basically for which $GL_{n+1}(F_p)$ does there exist an element of order $p^{n}$.
 
Still writing it, hit sumbit by mistake, bear with me.
 
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