Prove that a finite group of isometries cannot contain two halftums about distinct points.

8.4 Problems

1. Let S and T be two involutive transformations of the plane. (a) Prove that S Tis involutive if and only if S T = T S. (b) Assume that S, T, and I are distinct transformations, where I is the identity, such that ST=TS=X. Let r = {I, S, T, X}. Prove that r is a commutative subgroup of g, the group of all transformations on the plane, by constructing the multiplication table.

2. Let P, Q, and R be three points in the plane, and let P’, Q’, and R’, respectively, be their images under an isometry T. Show that the points P, Q, and Rare collinear, with Q between P and R, if and only if the points P’, Q’, and R’ are collinear, with Q’ between P’ and R’. Hint: When does equality hold in the Triangle Inequality?

3. LetT be an isometry of the plane. Show that if P and Q are fixed points ofT, then every point X on the line through P and Q is a fixed point ofT.

4. Let T be an isometry of the plane. Show that if T has three fixed points that are not collinear, then T =I, the identity.

5. LetS and T be isometries and let A, B, and C be three noncollinearpoints for which S(A) = T(A), S(B) = T(B), and S(C) = T(C). Show that S = T

. 6. Let H A be a halftum about a point A so that HA(P) = P’, where A, P, and P’ are collinear and d(A, P) = d(A, P’). p
A
(a) Show that HA is an isometry. (b) Show that HA is an involution; that is, HA = HA_1. (c) Show that if£ is a line in the plane, then H A ( £) is a line parallel to £.

Since RmRz is a member of Q, and since S contains all of the rotations in Q, it follows that RmRl = RQ,ka for some integer k. Therefore, we have RmRQ,ka = Rm (RmRz) = (RmRm) Rzm = Rz.
Leonardo’s Theorem now follows from Theorem 10.2.3 and Theorem 10.2.4.
One of the consequences of Leonardo’s Theorem is:
D
Theorem 10.2.5. The group of symmetries of a polygon in the plane is either a cyclic group or a dihedral group.
Proof. Given a vertex A of a polygon, together with an adjacent vertex B, any symmetry of the polygon must map A onto one of the vertices of the polygon, in which case there are at most two possible adjacent vertices that can be the image of B. In other words, there are only a finite number of symmetries. Leonardo’s Theorem now tells us that the group of symmetries is either a cyclic group or a dihedral group.
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10.3 Problems

  1. Prove that a finite group of isometries cannot contain two halftums about distinct points.
  2. 2. Prove that the set of all halftums and all translations forms a group. 3. Prove that if a triangle is invariant under a reflection, then the triangle must be isosceles. 4. Which of the following sets of transformations form a group, and which do not form a group? (a) All translations. (b) All reflections. (c) All glide reflections. (d) All rotations. (e) All direct isometries. (f) All opposite isometries.
    Leonard, I. E., Lewis, J. E., Liu, A. C. F., Tokarsky, G. W., & Leonard, I. E. (2014). Classical geometry : Euclidean, transformational, inversive, and projective. John Wiley & Sons, Incorporated. Created from gcu on 2021-11-25 06:12:49. Copyright © 2014. John Wiley & Sons, Incorporated. All rights reserved.
    284 SYMMETRY AND GROUPS
  3. If Ho, Ho2 = Ho2 Ho, = T, prove that T =I, the identity transformation.
  4. Find a plane figure P such that its group of symmetries equal (a) the cyclic group c2 of order 2, (b) the cyclic group C 1 of order 1.
  5. Find a plane figure P such that its group of symmetries equal (a) the dihedral group D2 of order 4, (b) the dihedral group D1 of order 2.
  6. Find the group of symmetries of each of the following figures.
    (a) (b)
  7. Let g be a group of isometries whose subgroup of translations is generated by TAB, where AB -1= 0. Prove that if R£ E Q, then either AB is parallel to Cor AB is perpendicular to C.
  8. Find the group of isometries of an ellipse. 11. If a and b are elements of a group g and
    show that ab = ba. 12. Let a and b be elements of a group g such that b has order 2 and ab = ba-l. (a) Show that anb = ba-n for all integers n. Hint: Evaluate the product (bab) (bab) in two different ways to show that ba2b = a-2, and then extend this method. (b) Show that the set S = {an, ban I n E Z} is closed under multiplication and in fact forms a group. (c) Show that S = (a, b), the dihedral group with generators a and b.
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