School of Mathematics and Statistics MT4516 Finite Mathematics
数学代写ASSIGNMENT Let P be the set of all 1-dimensional subspaces of F3 , and let L be the set of all 2-dimensional subspaces of F3.
1.Consider the construction of Theorem 3.5 of Chapter 4, which builds a fifinite affiffiffine plane (P,L) from n − 1 MOLS A1, . . . , An−1. 数学代写ASSIGNMENT
(a) Let V be the set of all vertical lines, let H be the set of all horizontal lines, and for each m ∈ {1, . . . , n − 1} let L(m) be the set of sloped lines for this fifixed m.
Prove that {V, H, L(1), . . . , L(n − 1)} form the parallel classes of L.
(b) Prove that the construction satisfifies the following properties:
(1) there are n2 points and n2 + n lines;
(2) every line contains n points;
(3) each point is contained in exactly n + 1 lines;
(4) any two distinct lines have at most one point in common.
2.Let (P,L) be a fifinite projective plane. Prove that if l is a line and if P is a point such that P ∉ l, then the number of points on l is equal to the number of lines through P.
Prove that there exists a positive integer m such that
(i) every point lies on exactly m + 1 lines and every line has exactly m + 1 points;
(ii) there are exactly m2 + m + 1 points and exactly m2 + m + 1 lines.
(Hint: Count the points and pairs of points in two difffferent ways each.)
3.Let F be a fifinite fifield, and consider the vector space F3. 数学代写ASSIGNMENT
Let P be the set of all 1-dimensional subspaces of F3 , and let L be the set of all 2-dimensional subspaces of F3 .
(i) Prove that (P,L) is a fifinite projective plane; denote this plane by PP(F). (Hint: dim(U + V) = dim(U) + dim(V) − dim(U ∩ V).) What is the number of points in PP(F)?
(ii) List the lines and the points of PP(Z2).
(iii) In PP(Z7) fifind: (a) the intersection of the lines {(x, y, z) : 2x + 3y + 4z = 0} and {(x, y, z) : 3x + 4y + 5z = 0}; 数学代写ASSIGNMENT
(b) the line containing the points {(t, 3t, 5t) : t ∈ Z7} and {(2t, 5t, t) : t ∈ Z7}.
4.Let O be a set of points in a fifinite projective plane of order n2 + n + 1 such that no three points of O lie on the same line.
(i) Prove that |O| ≤ n + 2. (Hint: Consider the lines through a point belonging to O.)
(ii) Prove that if |O| = n + 2 then every line that intersects O, intersects it in exactly two points.
(iii) Prove that if n is odd then |O| ≤ n + 1. (Hint: Consider the lines through a point not belonging to O.)
5.Does there exist:
(i) A fifinite projective plane of order 42?
(ii) A fifinite projective plane of order 57?
(iii) A fifinite projective plane of order 211?
(iv) A fifinite affiffiffine plane of order 100?
(v) A fifinite affiffiffine plane of order 196?
(vi) A fifinite affiffiffine plane of order 289?
