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- Let
*f*be the element*f*= (*ax*+*b*)/(*cx*-*a*) with*a*,*b*∈**R**,*a*^{2}+*bc*> 0. Find the order of*f*in the projective group*PGL*(2,**R**).

Let*g*be the element*g*= (*x*-1)/*x*and let*h*be*h*= 1 -*x*. Find the orders of*g*and*h*in*PGL*(2,**R**).

Calculate the elements of the subgroup of*PGL*(2,**R**) generated by*g*and*h*. What is the group? - Show that any three distinct points on a projective line can be mapped to any three distinct points on a second line by positioning the lines suitably in a plane and projecting from a point not on either line.
- Show that any four points (no three on a line) in
**R***P*^{2}can be mapped to any other four such points by a projective transformation.

[Hint: Show first that one can map [*α*, 0, 0], [0,*β*, 0] and [0, 0,*γ*] to three of the points. Then choose*α*,*β*,*γ*so that [1, 1, 1] is mapped to the fourth point.] - Given any vector space
*V*over a field*F*, we can form its associated projective space*P*(*V*) by using the construction from lectures:

*P*(*V*) =*V*- {0}/~ where ~ is the equivalence relation*u*~*v*if*u*=*λ**v*for*u*,*v*∈*V*- {0} and*λ*∈*F*.

Let*F*be a finite field with*p*^{k}elements where*p*is a prime number.

Prove that a*projective line*over*F*has*p*^{k}+ 1 points on it and that in general if*V*has dimension*n*over*F*, then |*P*(*V*)| = (*p*^{kn}- 1)/(*p*^{k}- 1).Deduce that if |

*F*| = 2 and*n*= 3 we get a*projective plane*with 7 points.

(The picture on the right represents such a plane. It has 7 points and 7 lines: each point lies on 3 lines and each line contains 3 points. One of the lines is represented by a circle.)Describe how one could construct projective planes with 13 points and with 21 points.

Solution to question 4

- Prove Pappus's theorem (about 320 AD):

If the vertices of a hexagon lie alternately on two lines, then the meets of opposite sides are collinear.[Hint: Project two of the meets to points on the line at infinity.]

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