Groups, Group Theory, Symmetry

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Groups, Group Theory, Symmetry

• A. D. Rollett
• 27-750, Spring 2003
• Advanced Characterization Microstructural
  Analysis

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Reference

• Kiral, E. and A. C. Eringen (1990). Review of
  group theory and representation. In Constitutive
  Equations of Nonlinear Electromagnetic-Elastic
  Crystals , Springer-Verlag, New York, pp. 183-209.

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Group Theory

• Groups are a very useful concept for dealing with
  symmetry because they allow one to collect
  together sets of symmetry operators.
• Group theory is a significant branch of
  mathematics, independent of symmetry.
• The Group Property is that property of a set of
  elements in which the associative combination of
  a pair of elements is equivalent to another
  member of the group.

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What is a Group, really?

• A group is (non-technical) a collection of
  quantities that form a closed system in which
  they are all related to one another.
• The closed nature of a group (set of elements)
  requires a set of rules (axioms) be written down
  that describe the relations (results of
  multiplication) between the elements of the group.

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Group Axioms (a.k.a. the rules)

• Definitions O ? element G ? group ?
  combination operator.
• 1. ? Oi,Oj ?G, OiOj Ok ?G (Group Property)
• 2. Oi(OjOk) (OiOj)Ok (Associativity)
• 3. ? I ?G, ? Oi ?G, I Oi Oi OiI. (Unit
  Element, I)
• 4. ? Oi-1 ?G, ? Oi ?G, Oi-1 Oi I Oi Oi-1.
  (Inverse Element, Oi-1)

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Other Properties of Groups

• Abelian Groups combination is commutative
  (reversible) OiOj OjOi this is not typical
  for symmetry groups because rotations do not
  commute.
• Order of a Group this is the number of elements
  in the group, e.g. 6 for point group 6.
• Cyclic groups all elements can be generated from
  powers of a single generating element. E.g.
  rotate by 2p/n

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Notation

• Crystallography operators are called n-fold axes
  (or just n on diagrams) following the
  International notation. The notation nm denotes
  the mth power of n e.g. 3 denotes rotation by
  2p/3, 32 denotes rotation by 4p/3. Subscript
  denotes the axis along which rotation occurs.
• Schoenflies operators are denoted by Cn
• Alternate notation Luvwn where uvw denotes the
  axis along which the symmetry element operates,
  and n denotes the axis order.

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Multiplication Tables for Groups

• The holosymmetric orthorhombic point group mmm

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Other Group Properties, contd.

• Homomorphism two groups are homomorphic if a
  unidirectional correspondence exists.
• Relation between higher lower order groups one
  or more elements from the higher order group can
  be inserted into the multiplication table of the
  lower group and still satisfy that table.
• Example the group of proper rotations
  represented by quaternions are homomorphic with
  but not the same as matrices.

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Other Group Properties, contd.

• Isomorphism special case of homomorphism two
  groups are isomorphous if a one-to-one
  correspondence (unique mapping) exists between
  the elements of the two groups.
• Example the group of proper rotations expressed
  as matrices is isomorphous with the group of
  Rodrigues vectors.

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Other Group Properties, contd.

• Subgroups if you can extract/identify a subset
  of elements within a group that also form a
  group, then that set is a subgroup.
• Improper subgroups I and the group itself.
• Proper subgroups subgroups other than above.
• Example out of 23, 2 is a subgroup, as is 3.

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Other Group Properties, contd.

• Cosets there are left cosets and right cosets
  the right coset of a subgroup is the set of
  elements formed by taking combinations of the
  subgroup with the remaining elements, writing the
  latter on the right.For a subgroup, H, of order
  p HRk H1Rk, H2Rk, .,HpRk

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Cosets, contd.

• Every element is either in the subgroup or one of
  its cosets.
• No element is in both a subgroup a coset.
• No element in more than one coset.
• No repeated elements.
• Order of a subgroup is a divisor of the order of
  the group (e.g. 2 and 3 divide 23).

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Representation

• Elements of groups commonly represented by
  matrices.
• Combination therefore represented by matrix
  multiplication.
• Proper Rotations (generally sufficient for
  representing orientation) are represented by the
  (infinite) group of orthogonal rotation matrices.
• Symmetry elements represented by unimodular
  matrices.

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Rotation Matrices
2x 180 about x-axis
Arbitrary rotationabout x-axis
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Table of Symmetry Groups
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Useful Groups

• O(3) all orthogonal matrices
• SO(3) all proper rotations (determinant1)
• O(432) cubic point group with proper rotations
• O(222) orthorhombic point group
• Cubic-orthorhombic symmetry represented by
  O(222)\g/O(432)

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Summary

• Group theory summarized, as it applies to
  symmetry for texture analysis.