**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 L_{1} and L_{2} are two lines in **L**, then
the intersection of L_{1} and L_{2} consists of exactly one
point.

(For example, one forms
a projective plane from the usual plane **R**^{2}
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 *S*_{7} of cardinality 168.

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