Course MT4521 Geometry and topology

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Exercises 4

  1. If A and B are two subgroups of I(R2) each generated by a rotation by 2π/n (about different points of R2), then prove that A and B are conjugate subgroups.
    (i.e. for some gI(R2) we have A = g-1Bg.)
    Deduce that although there are many different subgroups of I(R2) isomorphic to Cn they are all conjugate.

    Prove the same result for subgroups isomorphic to Dn.

    Prove that although the subgroups D1 and C2 are isomorphic as groups, they are not conjugate subgroups of I(R2).

    Solution to question 1

  2. A subgroup of symmetries of a set S is called discrete if there is some distance m such that every element of S is either fixed or moved by at least m by every element of G.
    If G is a discrete subgroup of I(R), prove that G contains a translation T by a minimum distance and that every translation in G is a power of T.

    Solution to question 2

  3. Prove that there are two infinite discrete subgroups of the group I(R), one generated by a translation (which we will call C iso Z under addition) and the other generated by a pair of reflections (which we will call D).

    Prove that the Frieze groups (i), ... , (vii) considered earlier are (respectively) isomorphic to C, C, D, D, D, C × D1, D × D1.

    Solution to question 3

  4. Inversion in a point aR3 or reflection in a is the map x goesto 2a - x.
    A rotatory inversion is rotation about a line L followed by inversion in a point of L.
    Show that every rotatory reflection is a rotatory inversion and vice versa.

    Solution to question 4

    1. Define a map θ from O(3) to the direct product SO(3) × {±1} by
      θ(A) = ( (det(A).A, det(A) ).
      i.e. θ(A) = ( A, 1 ) if ASO(3) and ( -A, -1 ) if ASO(3).
      Prove that θ is a group isomorphism.

    2. If X is a subset of R3 with -X = X (i.e. xX if and only if -xX) prove that the group S(X) of all symmetries of X is isomorphic to Sd(X) × < J > where Sd(X) is the subgroup of direct symmetries of X and < J > is the group of order 2 generated by J : x goesto -x.

    Solution to question 5

  5. Three identical cards with dimensions 2 × 2a for a a real number are assembled as shown with their centre lines along the axes. Write down the coordinates of the twelve corners of the cards.
    Show that if the number a is the Golden ratio φ = 1/2 (1+√5) then the distance AB is equal to the distance AC and deduce that the twelve points are at the vertices of an icosahedron.

    Solution to question 6

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JOC February 2010