I am unable to construct an affine plane of order $5$. I currently am able to construct an affine plane for orders $3$ and $4$ but I am not able to figure out the construction of an affine plane of order $5$ using a $5\times 5$ array of dots. Could someone show an example picture of one and how you were able to come to construct the affine plane of order $5$?
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There are enough lines that there is no nice elegant way to draw a picture of this. – xxxxxxxxx Nov 09 '20 at 15:06
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There is a field of order $5$, namely $\mathbb{F}_5 = \mathbb{Z}/(5)$, so you can just take the affine plane over $\mathbb{F}_5$.
The points are the elements of $\mathbb{F}_5\times \mathbb{F}_5$, and the lines are the sets of points of the form $\{(x,y)\mid y = ax + b\}$ and $\{(x,y)\mid x = c\}$ for $a,b,c\in \mathbb{F}_5$ (this is just the description of the affine plane over any field). There are $30$ lines, so a picture will inevitably get a bit muddled!
The point is that it's extremely easy to construct an affine plane of prime order $p$ in this way. Constructing an affine plane of prime power order $p^n$ is also straightforward - but a little more complicated to understand explicitly, because you first have to construct a field of order $p^n$ and understand its addition and multiplication tables: see here for example. On the other hand, the existence of affine planes of non-prime-power order is an open problem.
Alex Kruckman
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@Va1or If you are satisfied that my answer settles your question, you can accept it by clicking the green check mark. – Alex Kruckman Nov 09 '20 at 19:14