Illustrations of Simple Group Theory and the Dihedral Group

1. Illustrations of Simple Group Theory and the Dihedral Group

2. A little review…. The six rotations of an equilateral triangle form a group

3. (1) Leave the triangle alone (rotate 0°) (2) Rotate about OA (3) Rotate clockwise 120° (4) Rotate about OB (5) Rotate clockwise 240° (6) Rotate about OC

4. Produces no change – this is the identity operation (3) Produces a cyclic effect 1?3, 3?5, 5?1 On the reverse subset there is a reverse effect 2?6, 6?4, 4?2 (5) Produces a cyclic effect similar to that of (3) (2), (4), (6) are all reflexive operators producing the following:

5. Results can be tabulated using a table: It can be seen from the table that the group is not commutative (3) on 4 = 2 (4) on 3 = 6

6. The group of symmetries for an equilateral triangle is of order 6. Our book shows the group of symmetries for a square – this type of analysis can be done for any regular n-gon (n = 3). The corresponding group is denoted Dn and is called the dihedral group of order 2n.

7. If the order of Dn is greater than 4, the operations of rotation and reflection in general do not commute and Dn is not abelian. These are the simplest examples of non-abelian groups Generally, a finite set has 2n subsets where n is the size of the set. Note: generally, the order of a group does not determine the number of its subsets Thought: is there a simple way to express the number of subgroups of G, a finite group as a function of its parameters?

8. Theorem: The number of subgroups of the dihedral group Dn (n = 3), is d(n) + s(n) Note: d(n) is the number of positive divisors of n s(n) is the sum of the positive divisors of n

9. What are dihedral groups?

10. Examples D1 Shell gas uses this symbol. This shell shape has no rotations (other than the identity) and has only one mirror line (vertical). Therefore, like Mickey Mouse, the figure is said to be bilaterally symmetric and it fits into the category D1.

11. D2 An example of D2 that is easily spotted is the logo for the Central Broadcasting System (CBS). The "eye" shape within the circle prevents the figure from being able to rotate by any rotation other than a 1/2 turn. Additionally, the figure has only two ways in which it can be reflected onto itself.

12. D3 The luxury car, Mercedes-Benz, uses a symbol with three rotations and 3 mirror lines. Therefore, the emblem is an example of D3. If we were to convert this figure into a peace sign, however, we would lose 2 of the rotations and two of the reflection lines. This would leave a D1 figure.

13. D4 The symbol for Purina is a great example of a finite figure of the category D4. It is easy to see that there are four mirror reflections of the figure (one vertical, one horizontal, and two diagonal) as well as four rotations. In other words, rotating the figure four times gives the original figure (the identity).

14. D5 The symbol for Chrylser is a great example of a finite figure of the category D5. In other words, the symbol has five rotations and five axes of reflection.

15. D6 This finite figure is a dihedral group of order 8 due to its eight reflections and eight rotations. The symmetries are created by two squares placed on top of each other and offset by 90 degrees.

16. What other types of groups are there? Tetrahedral group – this is the group of order 12 . There are 3-fold axes (one through each vertex) and three 2-fold axes (joining the midpoints of non-interesecting edges.

17. The conjugacy classes of T are: identity 4 × rotation by 120° clockwise (seen from a vertex) 4 × rotation by 120° anti-clockwise 3 × rotation by 180°

18. The Icosahedron colored as a snub tetrahedron has chiral symmetry

19. Triakis tetrahedron can be seen as a tetrahedron with triangular pyramids added to each face. This interpretation is expressed in the name.

20. A tetrahedron can be placed in 12 distinct positions by rotation alone. These are illustrated above in the cycle graph format, along with the 180° edge (blue arrows) and 120° vertex (reddish arrows) rotations that permute the tetrahedron through those positions.

21. Octahedral group The rotation group of the cube that has order 24. For the cube, there are three 4-fold axes (joining centers of opposite faces), six 2-fold axes (joining midpoints of opposite edges), and four 3-fold axes.

22. The conjugacy classes of O are: identity 6 × rotation by 90° 8 × rotation by 120° 3 × rotation by 180° about a 4-fold axis 6 × rotation by 180° about a 2-fold axis

23. Achiral octahedral symmetry or full octahedral symmetry. This group has the same rotation axes as O, but with mirror planes, comprising both the mirror planes of Td and Th. This group is isomorphic to S4 × C2, and is the full symmetry group of the cube and octahedron

24. A dual cube-octahedron.

25. Icosahedral Group The rotation group of this has order 60. It has the same as the rotation group of the dodecahedron. There are six 5-fold axes (joining pairs of opposite vertices), ten 3-fold axes (joining centers of opposite faces), and fifteen 2-fold axes (joining the midpoints of opposite edges.

26. The conjugacy classesof I are: identity 12 × rotation by 72° 12 × rotation by 144° 20 × rotation by 120° 15 × rotation by 180°

27. In the disdyakis triacontahedron one full face is a fundamental domain