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Let $V=\mathbb{Z}_p\times \mathbb{Z}_p$, where $p$ is prime. Let $\pi:V\rightarrow V$ be a map such that if two lines are parallel in $V$, then their images remain parallel.

This is a property of affine transformations on $V$, so my question is whether it is sufficient, i.e. given that $\pi$ satisfies the property above are we guaranteed to have an invertible matrix $M\in \text{GL}(2,p)$ and $a\in V$ such that $$\pi(v)=Mv+a$$ for all $v\in V$?

JQX
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  • When you say "a cartesian product of finite fields" do you really mean "a vector space over a finite field"? (One might interpret a cartesian product of finite fields to be a ring, and then it doesn't make much sense to talk about "lines.") Also, certainly you can't guarantee that $M$ is invertible, since for example $\pi$ might be identically $0$. – Qiaochu Yuan May 20 '15 at 07:16
  • Also, are you also requiring that the image of a line remains a line? Otherwise it would be good to clarify what "parallel" means. – Qiaochu Yuan May 20 '15 at 07:19
  • Yes. That was sloppy on my part: what I meant was a two dimensional vector space $\mathbb{Z}_p\times \mathbb{Z}_p$. A line in this space is defined by the equation $ax+by=c$ for some $a,b,c\in \mathbb{Z}_p$. A point $(x,y)$ is in the line if it satisfies this equation. Two lines are parallel if their equations differ only at c. – JQX May 20 '15 at 19:03
  • Also yes, the map should preserve lines as well. – JQX May 20 '15 at 19:18

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