3P5 2000 Final Examination

All questions are of equal value.

 1. Let G be a group acting on a set X and let x be an element of X .

  1. Explain what is meant by the fixing group Gx of x and the orbit  orb(x) of x.
  2. Prove that Gx is a subgroup of G.
  3. Prove that there is a 1-1 correspondence between org(x) and the set [G:Gx] of right cosets of Gx in G.

2.

  1. Show that there is no simple group of order 20.
  2. A desk calendar is made by printing tables for each month on the faces of a regular dodecahedron, one month on each face. How many essentially different ways of doing this are there?

3. Let Iso(2) be the group of isometries of the Euclidean plane E2 and let ABC be a triangle in E2

  1. Show that if a, b in Iso(2) satisfy A a = A b , B a = B b and C a = C b then a = b, but if A a = A b and B a = B b, this is not necessarily the case..
  2. Explain the difference between direct and indirect isometries of E3.

4. Let O be the group of rigid motions of the cube.

  1. Explain why O is isomorphic to S4.
  2. Which elements of O correspond to A4 under this isomorphism?
  3. Explain why the group of symmetries of the cube has 48 elements.

5. Discuss in 10 lines or less for each part, each of the following:

  1. Sylow Theorems.
  2. Grieze groups.
  3. Linear representations of a group.
  4. The octahedral group.
  5. The action of conjugation on a group.

SOLUTIONS

1 (a) a = {a in G: xa = x}.

 orb(x) = { xa: a in G}.

 (b) Gx is not empty: since x1 = x, it contains the identity 1 of G.

 Let a, b in Gx. Since x ab = xb=x, ab  in Gx.

 Let a in Gx. Since x = x aa-1 = xa-1, a-1 in Gx.

(c) Define q : orb(x) -> [G:Gx] by q maps xa maps to Gxa .

q is 1-1: Gxa = Gxb implies a b-1 in Gx which in turn implies xa = xb .

 q is obviously onto.

2 (a) Let G be a group of order 20=22x 5.

Let N be the number of Sylow 5-subgroups. Then N divides 4 and 5 divides N-1. Hence N= 1.

Let P be the unique Sylow 5-subgroup of G. Then every conjugate of P is a Syulow 5-subgroup, so P is a proper normal subgroup. Hence G is not simple.

(b) There are 12! ways to print the calendars. The icosohedral group of order 60 acts on the set of ways to print the calendar. Only the identity fixes any of them, and it fixes all 12! hence by Not-Burnside's-Lemma, the number of distinguishable calendars is 12!/60.

3. (a) We must show that if P is in E2, then Pa = Pb. Now P is completely determined by its distances from A, B and C, so Pa is completely determined by its distances from the fixed points Aa, Ba and Ca and similarly Pb is completely determined by its distances from the fixed points Ab, Bb and Cb.

But |PaAa| = |PA| = |PaAb| = |PbAb| and similarly |PaBa| = |PB| = |PbBa| and |PaCa| = |PC| = |PbCa|. Hence Pa = Pb .

On the other hand, if only two of the distances are determined, there are two possible positions for Pa.

 (b) If a is a direct isometry of E3, then a preserves the orientation of all triangles.

If a is an indirect isometry of E3, then a reverses the orientation of all triangles.

4 (a) The four long diagonals of the cube are the only line segments in the cube of maximal length, hence O acts as a group of permutations of the long diagonals, creating a homomorphism of O into S4.

Checking each type of rigid motion shows that the kernel of the action is the identity. Since |O| = 24 = |S4|, the homomorphism is an isomorphism.

(b) The 8 rotations about the long diagonals realise the 3-cycles, and the 3 face axis rotations by p realise the products of two disjoint cycles. Together with the identity, this is all A4.

(c) There can be no more than 48 symmetries of the cube, since each must be direct or indirect, maps one of six faces to each fixed face, and one of four edges of this face to a given edge..

To realise them, take the 24 rigid motions and the product of the central inversion with each.

5 (a) Given a group G and a prime p, |G| = prm for some natural number r and some positive integer m prime to p. Then

  1. G has a subgroup P of order pr
  2. Every p-subgroup of G is contained in a conjugate of P.
  3. If P has N conjugates, then N divides m and p divides N-1.

(b) A Frieze group G is a discrete subgroup of Iso(2) containing a translation T such that every translation in G is a power of T. G is the symmetry group of a frieze, or linear pattern, and is one of seven types.

(c) If G is a group, a linear representation of G is a homomorphism of G into a group of linear transformations of a vector space.

(d) The octahedral group is the group of rigid motions of the cube, or the octahedron.

(e) Let G be a group, and define an action of G on G by xg = g-1 x g. This is a group action since for all x in G:

  1. For all g in G, g-1 x g is in G.
  2. x1 = x
  3. for all g, h in G, (xg)h = xgh.

Created June 18, 2000

Author: Phill Schultz, schultz@maths.uwa.edu.au