189-235A: Basic Algebra I

Assignment 10

Due: Wednesday, November 19.

1-2. Page 97, 2.49, 2.50.

3. A projective plane is a set P together with a collection L of distinguished subsets of P called lines. (The elements of P are sometimes called points.) These are subject to the following axioms:

i) If P and Q are two points in P, then there exists a unique line L in L containing P and Q.

ii) If L₁ and L₂ are two lines in L, then the intersection of L₁ and L₂ consists of exactly one point.
(For example, one forms a projective plane from the usual plane R² by adjoining to it a "point at infinity" for each possible direction of parallel lines.)

a) Show that the set of seven points P={1,2,3,4,5,6,7} together with the set of seven lines
L={ {1,2,4}, {1,3,7}, {4,6,7}, {1,5,6}, {2,5,7}, {3,4,5}, {2,3,6} }
forms a (finite) projective plane.

b) The symmetry group of a projective plane is the set of bijections from P to P which send lines to lines. Let G be the symmetry group of the projective plane of part a). Show that G is a subgroup of S₇ of cardinality 168.

4-7. Page 107, 2.55, 2.56, 2.58, 2.59.