Description
1. (i) Prove that GL2(R) is a group.
(ii) Prove that O2(R) is a group.
2. Consider the following isometries of R2. Let the elements of the Euclidean group
be represented by 3 ×3 matrices.
(i) Write down a matrix A which represents rotation by 2π/3 followed by translation by the vector (3, 0).
(ii) Find the point in R2 which is fixed by A.
(iii) Write down a matrix B which represents reflection in the y-axis followed by translation by the vector (4, 0).
(iv) Find the equation of the line in R2which is fixed by B.
3. Show, by the following two different methods, that there are exactly 432 auto-
morphisms of the affine plane A2(F3).
(i) Count the number of elements in Af f2(F3).
(ii) Count point-by-point how many possibilities exist for the image of each
point in an automorphism of A2(F3).
4. (i) How many matrices belong to the group GL3(F3)?
(ii) How many distinct elements belong to the group P GL3(F3)?
(iii) How many automorphisms does the projective plane P2(F3) admit?
