💎Mathematical Crystallography Unit 2 – Fundamentals of Group Theory

Key Concepts and Definitions

• Group consists of a set of elements and a binary operation that combines any two elements to form a third element in the set
• Binary operation must be associative, have an identity element, and each element must have an inverse
• Order of a group refers to the number of elements in the set
• Abelian group has the additional property that the binary operation is commutative (order of elements doesn't matter)
• Cyclic group can be generated by a single element (generator) through repeated application of the binary operation
• Isomorphism is a one-to-one correspondence between two groups that preserves the group operation
  • Isomorphic groups have the same structure and properties, even if they appear different
• Homomorphism is a function between two groups that preserves the group operation, but may not be one-to-one or onto

Group Axioms and Properties

• Closure axiom states that for any two elements $a$ and $b$ in a group $G$, the result of the binary operation $a * b$ is also in $G$
• Associative axiom requires that for any elements $a$, $b$, and $c$ in $G$, $(a * b) * c = a * (b * c)$
• Identity element axiom guarantees the existence of an element $e$ in $G$ such that for any element $a$ in $G$, $a * e = e * a = a$
• Inverse element axiom states that for each element $a$ in $G$, there exists an element $a^{-1}$ in $G$ such that $a * a^{-1} = a^{-1} * a = e$
• Commutativity is an additional property that some groups (Abelian groups) have, where for any elements $a$ and $b$ in $G$, $a * b = b * a$
• Cancellation law holds in groups, stating that if $a * b = a * c$ or $b * a = c * a$, then $b = c$
• Uniqueness of identity and inverses can be proven using the group axioms

Types of Groups

• Trivial group consists of only the identity element
• Cyclic groups are generated by a single element and can be represented as $\{e, a, a^2, \ldots, a^{n-1}\}$ where $a^n = e$
  • Examples include the additive group of integers modulo $n$ and rotational symmetry groups
• Dihedral groups ($D_n$) describe the symmetries of regular polygons, including rotations and reflections
  • $D_n$ has $2n$ elements and is non-Abelian for $n \geq 3$
• Symmetric groups ($S_n$) consist of all permutations of $n$ distinct objects
  • $S_n$ has $n!$ elements and is non-Abelian for $n \geq 3$
• Matrix groups are sets of invertible matrices with matrix multiplication as the binary operation (general linear group $GL(n, \mathbb{R})$)
• Point groups describe the symmetry of molecules and crystals, including rotations, reflections, and inversions

Subgroups and Cosets

• Subgroup is a subset of a group that forms a group under the same binary operation
  • Must contain the identity, inverse of each element, and be closed under the operation
• Proper subgroup is a subgroup that is not equal to the entire group
• Trivial subgroup consists of only the identity element and is a subgroup of every group
• Coset of a subgroup $H$ in a group $G$ is the set of elements obtained by multiplying each element of $H$ by a fixed element $a$ of $G$
  • Left coset: $aH = \{ah : h \in H\}$
  • Right coset: $Ha = \{ha : h \in H\}$
• Lagrange's theorem states that the order of a subgroup divides the order of the group
  • Consequence: the order of an element divides the order of the group
• Normal subgroup is a subgroup $N$ of $G$ such that $aN = Na$ for all $a$ in $G$ (left and right cosets are equal)
  • Normal subgroups are important for constructing quotient groups

Symmetry Operations in Crystallography

• Symmetry operation is a transformation that leaves an object appearing unchanged
• Translation symmetry involves shifting the object by a specific distance in a particular direction
  • Lattice is an infinite array of points generated by translational symmetry
• Rotational symmetry is characterized by the angle of rotation (e.g., 60°, 90°, 120°, 180°)
  • Rotation axis is the line around which the rotation occurs
• Reflection symmetry involves mirroring the object across a plane (mirror plane)
• Inversion symmetry is achieved by rotating the object 180° around a point (inversion center)
• Improper rotation combines a rotation with a reflection perpendicular to the rotation axis
• Glide reflection combines a reflection with a translation parallel to the mirror plane
• Screw axis involves a rotation followed by a translation along the rotation axis

Group Representations

• Representation of a group is a homomorphism from the group to a group of invertible matrices (matrix representation)
  • Captures the abstract structure of the group in a concrete form
• Degree of a representation is the dimension of the matrices used
• Faithful representation is a representation that is also an isomorphism (one-to-one correspondence)
• Irreducible representation cannot be decomposed into smaller matrix representations
  • Forms a basis for understanding the structure of the group
• Character of a representation is the trace (sum of diagonal elements) of the matrix corresponding to each group element
  • Characters can be used to analyze the properties of representations
• Great Orthogonality Theorem relates the characters of irreducible representations to the structure of the group
• Representation theory has applications in physics, chemistry, and materials science

Applications in Crystal Systems

• Crystal systems are defined by the symmetry elements present in the crystal structure
• Seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic
  • Each system has characteristic symmetry operations and point groups
• 230 space groups describe the possible symmetries of three-dimensional crystal structures
  • Combination of point group symmetry and translational symmetry
• International Tables for Crystallography provide a standard notation (Hermann-Mauguin notation) for describing space groups
• Point group determines the anisotropy of physical properties (e.g., optical, electrical, magnetic)
• Space group places constraints on the atomic positions and chemical composition of the crystal
• Group-subgroup relationships can be used to understand phase transitions and structural relationships between crystals

Problem-Solving Techniques

• Cayley table is a square table that shows the result of the binary operation for each pair of group elements
  • Helps to verify group axioms and identify properties like commutativity
• Cycle notation is a compact way to represent permutations in symmetric groups
  • Useful for understanding the structure and order of permutations
• Symmetry elements can be identified by looking for invariance under transformations
  • Flow chart approach: systematically check for the presence of each type of symmetry element
• Multiplication table for a point group lists the products of all pairs of symmetry operations
  • Used to determine the group structure and identify subgroups
• Irreducible representations can be found using the character table and the Great Orthogonality Theorem
  • Decompose a reducible representation into a sum of irreducible representations
• Symmetry-adapted linear combinations (SALCs) are basis functions that transform according to irreducible representations
  • Used in the application of group theory to molecular orbitals and vibrational modes
• Compatibility relations determine the symmetry-allowed transitions and couplings between states or orbitals
  • Selection rules based on the direct product of irreducible representations