Course MT3818 Topics in Geometry

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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 CZ 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 ↦ 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 A notbelongs SO(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 ↦ -x.

    Solution to question 5

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JOC March 2003