3P5 1999 Final Examination

All questions are of equal value.

 1. Let G be a group acting on a set X and let S be a subset of X . Let

GS = {g in G: xg = x for all x in S}.

  1. Prove that GS is a subgroup of G.
  2. How many different actions are there of S2 on {1,2,3}?
  3. Explain why no action of S2 on {1,2,3} can be transitive.

2.

  1. Show that any group of order 36 has a normal subgroup of order 3 or 9.
  2. How many essentially different necklaces can be formed from 2 red, 2 white and 2 blue beads?

3. Let Iso(2) be the group of isometries of the Euclidean plane E2

  1. Show that every element of Iso(2) is the product of at most 3 reflections.
  2. Explain what is meant by a frieze group.

4. Let T be a regular tetrahedron and let G be its group of symmetries.

  1. Show that G is isomorphic to S4.
  2. Which elements of G correspond to products of two transpositions under this isomorphism?
  3. Let E and F be opposite edges of T. Note that the midpoints of the other four edges are the corners of a square. (No need to prove this). Let A and B be the two pieces into which T is divided by the plane containing this square. Find the element of S4 whose corresponding symmetry of T maps A onto B.

5. Explain in 10 lines or less for each part what is meant by the following:

  1. the crystallographic restriction.
  2. the wallpaper groups.
  3. the length of an orbit is the index of the fixed group.
  4. the regular icosohedron.
  5. the action of conjugation on a group.

SOLUTIONS

1 (a) Let g, h in GS. Then for all x in S, x = xgg-1 = x g-1 so g-1 in GS, and for all x in S, xgh = (xg) h = xh = x so gh in GS. Hence GS is a subgroup of G.

 (b) There is a 1--1 correspondence between actions of S2 on {1,2,3} and homomorphisms f:S2 into S3. Since every f maps identity to identity, any such homomorphism is determined by the image of (1,2) , which must be an element of order 1 or 2.

There are four such homomorphisms, f0 mapping (1,2) to the identity, f1 mapping (1,2) to (1,2) , f2 mapping (1,2) to to (1,3) , and f3 mapping (1,2) to (2,3)

 (c) An action of S2 on {1,2,3} is transitive if and only if it has an orbit of length 3. But the length of an orbit divides the order of the group, so no action can be transitive.

2 (a) Let G be a group of order 36=2232. Let G have n3 3--Sylow subgroups which are of order 9. Then n3 divides 4 and 3 divides n3-1, so n3 = 1 or 4.

 If n3 = 1, then G has a normal subgroup of order 9.

 If not, G has 4 conjugate subgroups A, B, C and D each of order 9. The intersection of each pair has order 3 so all these intersections are equal, and hence normal. Therefore G has a normal subgroup of order 3.

 (b) The number of colourings of a necklace of 2 white, 2 red and 2 blue beads is (6 choose 2).(4 choose 2) = 90.

By Not Burnside's Lemma, the number of essentially different necklaces is the number of orbits of D = D12 acting on the set of colourings =1/|G|(Sg in D|Xg|).

 The identity fixes 90 colourings.

 Let r be rotation by p/3. r, r2, r4 and r5 fix no colourings.

r3 fixes every colouring which has antipodal beads of the same colour. There are 6 of these.

 Let f be reflection in an axis through two beads. There are 3 of these, and each fixes colourings which have axis beads of the same colour and opposite beads of the same colour. There are 6 such colourings. Hence reflections of this type fix 18 colourings.

Let t be a reflection in an axis between two beads. There are 3 of these, and each fixes a colouring which has opposite beads of the same colour. There are 6 of these. Hence reflections of this type fix 18 colourings.

 Thus the sum of the fixed sets has size 132 so the number of essentially different necklaces is 132/12=11.

3. See notes

4 (a) G has 12 direct symmetries, consisting of the identity, the 8 rotations by 2 p/3$ and 42 p/3 about axes through a vertex and the centroid of the opposite side, and the 3 half turns about the axes joining midpoints of opposite sides; and 12 indirect symmetries, consisting of the products of each of these with a reflection in a plane through a vertex and an altitude of the opposite side.

 Label the vertices {1,2,3,4} and associate each symmetry with its action on this set. This gives an isomorphism of T onto S4.

 (b) There are 3 types of products of two transpositions, (12)(12), (12)(34) and (12)(13) . All such products are conjugate in S4 to one of these. The first is associated with the the identity symmetry, the second with the half turn through the axis joing the mid-point of 12 to the midpoint of 34 and the third is associated with the rotation by 2S4/3 or 4S4/3 about the axis through vertex 4.

 (c) We are looking for a symmetry of order 2 which maps one pair of vertices to the complementary pair. Hence the symmatry is (12)(34) (or one of its conjugates).

5 (a) See notes

 (b) See notes

 (c) Let G be a group acting on a set X. If x in X , then orbit (x) {xg: g in G } . The fixed group of x, Gx = {g in G: xg = x }. Now Gx < G so [G:Gx] is the set of cosets of Gx in G. The index of Gx in G is |[G:Gx]|.

The map xg mapsto Gxg is a 1--1 correspondence of orbit (x) onto [G:G_x].

 (d) The regular icosohedron is the Platonic solid having 12 vertices and 20 equilateral triangle faces, 5 arranged round each vertex, and 30 edges. It is the dual of the regular dodecahedron. Its group of rigid motions is isomorphic to A5.

 (e) Consider a group G acting on itself by the action xg = g-1xg . This is a group action since x1 = x for all x in G and x{gh} = (xg )h for all x, g and h in G . The orbits are the conjugacy classes, the fixed groups are the centralizers.

 Created May 11, 2000

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