Illustrations of Simple Group Theory and the Dihedral Group

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Illustrations of Simple Group Theory and the
Dihedral Group
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A little review.

• The six rotations of an equilateral triangle form
  a group

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• (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

• To understand the structure of the group
  observe the effect of each operator on the
  complete set of orientations

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• Produces no change this is the identity
  operation
• (3) Produces a cyclic effect 13 35 51
• On the reverse subset there is a reverse effect
  26 64 42
• (5) Produces a cyclic effect similar to that of
  (3)
• (2) (4) (6) are all reflexive operators
  producing the following

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• 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

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• 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.

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• 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

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• 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

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What are dihedral groups
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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.

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• 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.

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• 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.

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• 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).

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• 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.

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• 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.

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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.

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• 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

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• The Icosahedron colored as a snub tetrahedron has
  chiral symmetry

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

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• 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.

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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.

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• 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

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• 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

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• A dual cube-octahedron.

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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.

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• The conjugacy classesof I are
• identity
• 12 rotation by 72
• 12 rotation by 144
• 20 rotation by 120
• 15 rotation by 180

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• In the disdyakis triacontahedron one full face is
  a fundamental domain

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