Course MT3818 Topics in Geometry

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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) ?

2. A plane lattice is called isosceles if it is of the form { ae1 + be2 | a, bZ } 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.

3. Let L be the lattice {ae1 + be2 | a,bZ } in R2. If gI(R2) is a symmetry which maps L to itself, prove that the matrix Mg of g with respect to the basis {e1 , e2} has integer entries.
Is Mg 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).

4. Let U, V be subgroups of a group G with UV = G. That is, every element gG can be written as g = uv with uU and vV. Prove that if UV = {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 UU1 , VV1 , UV = G and UV = {1}.

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

5. Let U be the subgroup < (123) > and let V be the subgroup < (12) > of G = S3 . Prove that UV = G and UV = {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.

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JOC March 2003