A finite projective plane of order \(n\) (\(n\geq 2\)) is a model that satisfies the Axioms Axiom 2.2.4, Axiom 2.2.5, Axiom 2.2.6, Axiom 2.2.20 and the following additional Axioms:
Axiom2.4.2.
Every pair of distinct lines intersect at at least one point.
Axiom2.4.3.
There is at least one line \(L\) containing exactly \(n+1\) points.
For the following proofs, some of them are really just rewording of known axioms/results. Some of them are more involved.
Problem2.4.9.
Show that for each point \(P\text{,}\) there are at least two lines which contain \(P\text{.}\)
Problem2.4.10.
Show that for each point \(P\text{,}\) there is least one line which does not contain \(P\text{.}\)
Problem2.4.11.
Show that there is at least one point \(P\text{.}\)
Problem2.4.12.
Given any two lines \(L, M\text{,}\) they intersect at exactly one point.
Problem2.4.13.
Given any two points \(P, Q\text{,}\) there is at least one line containing them both.
Problem2.4.14.
There is at least one point \(P\) contained by exactly \(n+1\) lines.
Hint.
Hint.
By Axiom 2.4.3 there is a line \(L\) that has \(n+1\) points. Label these points \(P_1, P_2,\ldots, P_{n+1}\text{.}\)
Hint.
Explain why there must be a point \(A\) that is not contained by \(L\text{.}\) (What if every point was contained by \(L\text{?}\) What bad thing would happen?).
Hint.
Explain why for each \(P_i\text{,}\) there is a line containing it and \(A\text{.}\)
Hint.
Can we show that none of the lines we just found can be the same line? (What if \(\overleftrightarrow{P_i A}=\overleftrightarrow{P_i B}\text{?}\) What bad thing would happen?)
Hint.
How do we know these are all the lines which contain \(A\text{?}\) (Let \(M\) be some line containing \(A\text{.}\) Why does \(M\) have to be one of the lines we just found?)
Hint.
Conclude that \(A\) is (one of) the point(s) which is contained by exactly \(n+1\) lines.
What does all this tell you about the relationship between points and lines in finite projective planes?
Subsection2.4.2Counting points and lines
Problem2.4.16.
If a finite projective plane has \(k\) points, how many lines do you suppose it has? Why?
Problem2.4.17.
Show that a finite projective plane has at least one collection of 4 lines, no three of which intersect at the same point.
Problem2.4.18.
Show that a finite projective plane has at least one collection of 4 points, no three of which line on the same line. (You can show this directly, or what does the discussion in Problem 2.4.15 tell us?)
Problem2.4.19.
Given any line \(L\) containing \(x\) points, and any point \(A\) not contained in \(L\text{,}\)\(A\) is contained in exactly \(x\) lines. (Can we repurpose parts of the proof of Problem 2.4.14?)
Problem2.4.20.
Given any point \(P\) contained in \(x\) lines, and any line \(M\) not containing \(P\text{,}\)\(M\) contains exactly \(x\) points.
Problem2.4.21.
Show that given any two points \(A, B\text{,}\) there is a line containing neither \(A\) nor \(B\text{.}\)
Problem2.4.22.
Show that given any two lines \(L, M\text{,}\) there is a point contained in neither \(L\) nor \(M\text{.}\)
Problem2.4.23.
Show that if \(L\) is any line containing \(n+1\) points, and \(P\) is a point not contained in \(L\text{,}\) then \(P\) is contained in exactly \(n+1\) lines.
Problem2.4.24.
Show that if \(P\) is any point contained in \(n+1\) lines, and \(L\) is a line not containing \(P\text{,}\) then \(L\) is contains exactly \(n+1\) points.
Problem2.4.25.
Show that for every point in a finite projective plane, there exactly \(n+1\) lines containing that point.
Hint.
Hint.
There is a line \(L\) with \(n+1\) points. Why? Call these points \(P_1, P_2, \ldots, P_{n+1}\text{.}\)
Hint.
Let \(P\) be a point not on \(L\text{,}\) why are there exactly \(n+1\) lines containing \(P\text{?}\) See Problem 2.4.19.
Hint.
Now suppose instead that \(P\) were on \(L\text{.}\) There must be some \(Q\) not on \(L\text{.}\) Why?
Hint.
Show that there is a line \(M\) that contains neither \(P\) nor \(Q\text{.}\)
Hint.
\(Q\) is contained in exactly \(n+1\) lines, why? Call these lines \(L_1, L_2, \ldots, L_{n+1}\text{.}\)
Hint.
Do these lines \(L_i\) intersect \(L_i\text{?}\) Why or why not? Can two different lines \(L_i, L_j\) intersect \(M\) at the same point? Why or why not? Call the point where \(L_i\) intersects \(M\)\(A_i\text{.}\)
Hint.
Does \(M\) contain any other points besides the \(A_i\text{?}\) Why or why not? How many points does \(M\) have?
Hint.
So FINALLY, why is \(P\) contained in exactly \(n+1\) lines?
Problem2.4.26.
Show that every line contains \(n+1\) points.
Problem2.4.27.
How many points does a finite projective plane of order \(n\) have?
Problem2.4.28.
How many lines does a finite projective plane of order \(n\) have?
Problem2.4.29.
What order is the projective plane found in Problem 2.2.30? Verify that it has the right number of points and lines.
Problem2.4.30.
What order is the projective plane found in Activity 2.1.6? Verify that it has the right number of points and lines.
