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) ?

    Solution to question 1

  2. A plane lattice is called isosceles if it is of the form { ae1 + be2 | a, b belongs 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 belongs Z } in R2. If g belongs I(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).

    Solution to question 3

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

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

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

    Solution to question 5

SOLUTIONS TO WHOLE SET
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JOC March 2003