Advanced Geometry Exercises

1. Prove that any two 2-dimensional lattices are isomorphic as groups.
   Under what conditions are these groups conjugate subgroups of I(R²) ?

   Solution to question 1

2. A plane lattice is called isosceles if it is of the form { ae₁ + be₂ | a, b Z } where e₁ and e₂ are vectors of the same length.
   Prove that a lattice can have a reflection or glide as a symmetry if and only if it is either rectangular or isosceles.

   Solution to question 2

3. Let L be the lattice {ae₁ + be₂ | a,b Z } in R². If g I(R²) is a symmetry which maps L to itself, prove that the matrix M_g of g with respect to the basis {e₁ , e₂} has integer entries.
   Is M_g an orthogonal matrix ?
   Use the fact that the trace (the sum of the diagonal entries) of a matrix is left fixed by any change of basis to prove the crystallographic restriction (Rotations which map a lattice to itself can only have orders 1, 2, 3, 4 or 6).

   Solution to question 3

4. Let U, V be subgroups of a group G with UV = G. That is, every element g G can be written as g = uv with u U and v V. Prove that if U V = {1} then the terms in this product are uniquely determined.

   If a group G is isomorphic to a direct product U₁ V₁ prove that there are normal subgroups U, V of G with U U₁ , V V₁ , UV = G and U V = {1}.

   Prove conversely that if there are normal subgroups U, V of G with UV = G and U V = {1} then G is isomorphic to a direct product U V.

   Solution to question 4

5. Let U be the subgroup < (123) > and let V be the subgroup < (12) > of G = S₃ . Prove that UV = G and U V = {1} but that G is not a direct product of U and V.

   Prove that the group S₅ is not a direct product of A₅ and any subgroup of order 2.

   Let T be the subgroup of translations in I(Rⁿ) and let O(n) be the subgroup of linear symmetries. Prove that I(Rⁿ) is not a direct product of T and O(n) for n 1.

   Solution to question 5