Group theory
for physics students
Péter Bántay
Institute for Theoretical Physics, ELTE
Introduction
Group theory is currently one of the most important mathematical dis-
ciplines, with manifold applications in
· mathematics (Galois theory, di�erential equations, geometry, etc.)
· physics (crystallography, solid state physics, high energy physics,
gauge theories, phase transitions, general relativity, etc.)
· chemistry
· esoteric sciences (like economics, linguistics, etc.)
Question: why groups?
Origins of group concept: symmetry and (co)homology.
Homology is a topological notion characterizing the �connectedness� of
a manifold through a sequence of groups.
Has important physical applications in the study of quantum systems
(Berry phase), gauge theories (instantons), general relativity (gravita-
tional singularities) and string theory.
Can be generalized to more general situations (homological algebra).
Symmetry: invariance under suitable transformations.
Examples: bilateral (mirror) symmetry of the human body (characteris-
tic of a very large class of animals, the Bilateralia) to be contrasted with
the �ve-fold rotational symmetry of echinoderms and the icosahedral
symmetry of certain micro-organisms.
Similar symmetry patterns show up in arti�cial (man made) objects like
buildings, furniture, decorations, etc.
Groups describe the algebra of symmetries!
Group theory provides techniques to convert qualitative information into
quantitative one.
Examples:
1. Counting of phenomenological constants
σij = Lijklukl Hooke's law
with uij and σij denoting the deformation and the stress tensor.
Onsager's symmetry relations
Lijkl = Ljikl = Lklij = · · ·
Question: number of independent elastic coe�cients Lijkl?
Answer: 2 in an isotropic medium (the elastic moduli), 21 in a triclinic
crystal.
For a crystal with point group G (in any dimension)
1
8
∑g∈G
{Tr(g)
4+ 2Tr(g)
2Tr(g2)
+ 3Tr(g2)2
+ 2Tr(g4)}
More generally, the free energy (a scalar quantity) is an invariant poly-
nomial of the (symmetric) deformation tensor, hence the answer equals
the number of independent invariants of the symmetrized square of the
point group (above formula gives the number of quadratic invariants,
appropriate in the linear case).
2. Reduction of partial di�erential equations
The solutions of a boundary value problem with given symmetry group
can only depend on the invariants of that group solution may be
reduced to a problem with fever independent variables, by introducing
some of the invariants as new variables (the solution will only depend on
these).
As an example, consider a spherically symmetric potential V (r). Then
Schrödinger's equation for the wave function ψ(r, φ, ϑ)
− ~2
2m∆ψ + V ψ = Eψ
(describing the motion of a point particle of mass m) is rotationally
invariant, hence in case of symmetric boundary conditions (that are in-
dependent of the angle coordinates r, e.g. ψ(R,φ, ϑ) =ψ0 for some R)
its solution is independent of φ and ϑ, and the problem is reduced to
solving the ODE
1
r2d
dr
(r2dψ
dr
)+ V (r)ψ = Eψ
with boundary condition ψ(R)=ψ0.
Remark. Usually, because the boundary conditions can break some
symmetries of the equation, not all of them play a role (unlike in the
previous example).
1 HISTORICAL HIGHLIGHTS
1 Historical highlights
Antiquity: application of symmetry principles in geometry (Euclid of
Alexandria, Archimedes of Syracuse), classi�cation of platonic solids.
J.-L. Lagrange (1771): solubility of polynomial equations.
É. Galois (1832): Galois theory.
1 HISTORICAL HIGHLIGHTS
A. Cayley (1854): abstract group concept.
É. Mathieu (1861,1873): Mathieu groups.
F. Klein (1873): classi�cation of geometries (Erlangenprogram).
S. Lie (1871-1893): continuous transformation groups.
1 HISTORICAL HIGHLIGHTS
H. Poincaré (1882): homology groups, uniformization.
É. Picard (1883): di�erential Galois theory.
D. Hilbert (1888): invariant theory, homological algebra.
1 HISTORICAL HIGHLIGHTS
W. Killing (1880-1890), É. Cartan (1894): classi�cation of simple Lie-
algebras.
G.F. Frobenius (1896): representation theory, group char-
acters.
W. Burnside (1903): �nite groups.
1 HISTORICAL HIGHLIGHTS
I. Schur (1904): projective representations.
A. Haar (1933): invariant integrals.
1966-1976: classi�cation of �nite simple groups.
1 HISTORICAL HIGHLIGHTS
Group theory in physics
Renaissance: symmetry principles in statics (�Epitaph of Stevinus�).
L. Euler (1765): movement of rigid bodies.
E.S. Fedorov (1891), L. Schön�ies (1891) és W. Barlow
(1894): classi�cation of crystal structures in 3D.
H. Poincaré (1900): symmetries of Maxwell's equations.
1 HISTORICAL HIGHLIGHTS
E. Noether (1915): symmetries and conservation laws.
E. Wigner: symmetries in quantum physics (1933),
relativistic wave equations (1947).
C.N. Yang és R. Mills (1954): gauge theories.
M. Gell-Mann (1963): �eightfold way� (leading to the quark model).
D. Shechtman (1982): quasi-crystals.
2 FUNDAMENTAL CONCEPTS
2 Fundamental concepts
Q: What is a group? How to compare groups?
A group is a set of elements with a suitable binary operation.
Binary operation: rule that assigns to two elements of a set a well-de�ned
third element of the same set.
Examples: addition and multiplication of (integer, rational, real, com-
plex, hypercomplex, p-adic, etc.) numbers, greatest common divisor and
lowest common multiple of integers, addition and cross product of vec-
tors (but not their scalar product), addition and multiplication of linear
operators, polynomials, matrices, ...
2 FUNDAMENTAL CONCEPTS
Multiplicative in�x notation: for a binary operation on the set X, we
denote by x?y (or simply by xy) the element obtained by applying the
operation to the elements x, y∈X, and call it their product.
A binary operation (on the set X) is called
associative, if for any x, y, z∈X
x(yz) = (xy)z
commutative, if for any x, y∈X
xy = yx
unital, if there exists 1X∈X (the identity) such that for all x∈X
1Xx = x1X = x
2 FUNDAMENTAL CONCEPTS
A group is a set G of elements together with an associative and unital
binary operation (the 'product'), such that for all x ∈ G there exists
x-1∈G (the inverse of x) for which
xx−1 = x−1x = 1G
A homomorphism is a mapping φ :G1→G2 such that for all x, y∈G1
φ(xy)=φ(x)φ(y)
An isomorphism is a bijective (i.e. one-to-one) homomorphism.
The groups G and H are isomorphic, denoted G∼=H, if there exists an
isomorphism φ :G→H.
2 FUNDAMENTAL CONCEPTS
Re�exive (G ∼= G), symmetric (G ∼= H implies H ∼= G) and transitive
(G∼=H and H∼=K implies G∼=K) relation.
Isomorphy principle: isomorphic groups cannot be distinguished alge-
braically.
Automorphism: isomorphism α : G→ G of a group with itself. Auto-
morphisms form a group, denoted Aut(G), with composition as binary
operation (symmetries of algebraic structure).
Order of a group: cardinality∣∣G∣∣ of its set of elements.
The orders of isomorphic groups coincide: G∼=H ∣∣G∣∣= ∣∣H∣∣.
Abelian group: group product is commutative (very special properties).
2 FUNDAMENTAL CONCEPTS
Generalizations of the group concept:
· relaxing the existence of inverses leads to monoids (with applica-
tions to automata theory & linguistics, renormalization, etc.);
· relaxing the associativity of the product results in quasi-groups
(combinatorial applications like latin squares, aka. sudoku);
· a partially de�ned product gives groupoids (with applications to
topology, the description of quasi-crystals, etc.);
· theoretical physics quantum groups, supergroups, ...
2 FUNDAMENTAL CONCEPTS
Recommended reading:
A.A. Kirillov : Elements of the theory of representations, Springer (1976).
J.L. Alperin and B. Bell : Groups and representations, Springer (1995).
D. Robinson : A course in the theory of groups, Springer (1993).
W. Magnus, A. Karrass and D. Solitar: Combinatorial group theory,
Interscience Publishers (1966).
3 EXAMPLES
3 Examples
3.1 Additive and unit groups of rings
Consider a (unital) ring R , i.e. a set with two associative and unital
binary operations, addition ('+') and multiplication ('·'), that satisfy
1. addition is commutative, and each a ∈R has an additive inverse,
denoted −a, such that a+(−a)=(−a)+a=0, where 0 denotes the
additive identity (the zero element of R);
2. the distributive laws a (b+c)=ab+ac and (a+b) c=ac+bc hold for
any a, b, c∈R.
3 EXAMPLES
Examples of rings:
1. The sets Z, Q, R, C of integers, rationals, reals, resp. complex
numbers, with the usual addition and multiplication are commu-
tative rings (but the set 2Z+1 of odd integers is not a ring, since
the sum of two odd numbers is even).
2. The set Z/nZ = {nZ+k | 0≤k
3 EXAMPLES
3. For a commutative ringR and a setX, the set ofR-valued functions
on X (i.e. maps from X to R) with pointwise operations
(f + g)(x) = f(x) + g(x)
(fg)(x) = f(x) g(x)
for x∈X, is again a commutative ring, denoted usually R(X). If
both R and X have some kind of geometric structure (e.g. topo-
logical, di�erentiable, complex analytic, etc.), then the structure
preserving (i.e. continuous, di�erentiable, holomorphic, etc.) maps
form a ring themselves.
3 EXAMPLES
4. For a commutative ring R and a positive integer n, a polynomial
in n indeterminates is a map from Zn+ into R with �nite support,
i.e. which takes on non-zero values at only a �nite number of
arguments. The addition of polynomials is de�ned pointwise, while
the product of the polynomials P and Q reads
(PQ)(α1, . . . , αn) =∑
0≤βi≤αi
P (β1, . . . , βn)Q(α1−β1, . . . , αn−βn)
With these operations, polynomials form the ring R[x1, . . . , xn].
The indeterminates (generating the whole ring) are de�ned as the
polynomials xi(α1, . . . , αn)=δαi,1∏j 6=i
δαj ,0.
3 EXAMPLES
5. For two positive integers n and m and a commutative ring R, the
maps from the set X = {1, . . . , n}×{1, . . . ,m} into R are called
n-by-m matrices over R, and their set is denoted Matn,m(R); a
matrix is square of size n if m = n. The values of the matrix
A ∈ Matn,m(R) are called its matrix elements, and are denoted
Aij = A(i, j) for 1 ≤ i ≤ n and 1 ≤ j ≤ m. The addition of the
matrices A ∈Matn,m(R) and B ∈Matp,q(R) is de�ned pointwise,
(A+B)ij=Aij+Bij , provided p=n and q=m, while their product
is de�ned when p=m, in which case it reads (AB)ij=m∑k=1
AikBkj .
It follows that the set Matn(R) of square matrices of any size n
forms a ring, which is not commutative unless n=1.
3 EXAMPLES
The set of elements of a ring R with the operation of addition forms an
Abelian group, the additive group of R.
A unit is an element a∈R having a multiplicative inverse a−1 such that
aa−1 = a−1a = 1R
denoting by 1R the multiplicative identity ('one') of R.
The set of units with the operation of multiplication forms a group R×,
the unit group of R.
A �eld F (like Q, R or C) is a ring whose unit group contains all its
nonzero elements: F×=F \{0}.
3 EXAMPLES
Examples:
1. Z×={±1};
2. (Z/nZ)× = {nZ+k | 0≤k0.
5. Matn(R)×
= GLn(R) = {A∈Matn(R) | detA∈R×}, the general
linear group over R, consisting of the invertible n-by-n matrices.
3 EXAMPLES
3.2 Matrix groups
Besides the general linear group GLn(R), several other subsets of Matn(R)
form a group with the operation of matrix multiplication:
· the set∆n(R) = {A∈GLn(R) |Aij=0 if i 6=j}
of diagonal matrices is an Abelian group;
· the setMn(R) of monomial matrices, in which each row and column
contains exactly one nonzero entry;
3 EXAMPLES
· the set
Πn = {A∈Mn(R) |Aij 6=0 implies Aij=1R}
of permutation matrices (note that every monomial matrix is the
product of a diagonal and a permutation matrix);
· the setOn(R) =
{A∈GLn(R) |A−1 =A
ᵀ}of orthogonal matrices (where A
ᵀdenotes the transpose of a ma-
trix A∈Matn(R), with entries (Aᵀ)ij = Aji), and more generally
Op,q(R) ={A∈GLp+q(R) |A−1 =η−1p,qA
ᵀηp,q
}with ηp,q denoting the diagonal matrix having the �rst p diagonal
entries equal to 1R, and the remaining q entries equal to −1R;
3 EXAMPLES
· the set
Sp2n(R) ={A∈GL2n(R) |A−1 =J−1n A
ᵀJn
}of symplectic matrices, where Jn is the block-diagonal matrix made
up of n copies of the Pauli-matrix
iσ2 =
(0 1R
−1R 0
)
· subsets of all the above whose elements satisfy the requirementdetA=1R, e.g. the special linear and orthogonal groups
SLn(R) = {A∈GLn(R) | detA=1R}
SOn(R) = {A∈On(R) | detA=1R}
3 EXAMPLES
3.3 Symmetric and alternating groups
Permutation: bijective self-map of a (�nite) set into itself.
Product of permutations: composition of bijective maps (associative op-
eration, with the trivial permutation idX:x 7→x as identity element).
Inverse of a permutation: inverse map (again bijective).
The collection of all permutations of a �nite setX forms a group Sym(X),
the symmetric (NOT symmetry) group over X (not commutative in case
|X|>2), of order ∣∣Sym(X)∣∣ = ∣∣X∣∣!Transposition: interchange of two elements.
3 EXAMPLES
Any permutation can be decomposed (in many ways) into a product of
transpositions: while their number is not unique, their parity (whether
there is an even or odd number) characterizes the permutation odd
and even permutations.
The product of even permutations is even even permutations form
themselves a group, the alternating group Alt(X) (of order half of that
of Sym(X)).
Sym(X) and Sym(Y ) (resp. Alt(X) and Alt(Y )) are isomorphic exactly
when∣∣X∣∣= ∣∣Y ∣∣
Sn=Sym({1, . . . , n}) and An=Alt({1, . . . , n}), the symmetric (alternat-
ing) groups of degree n (isomorphic to Πn and SΠn respectively).
3 EXAMPLES
3.4 Geometric symmetry groups
Symmetries of geometric �gure: rigid motions (Euclidean isometries)
mapping the �gure (as a set of points) onto itself.
Regular polygon: convex plane �gure all of whose sides are congruent
(i.e. have equal length) angles between neighboring sides are also
equal.
Up to similarity, exactly one regular n-gon for each integer n>2.
3 EXAMPLES
Medians (edge bisectors) of a regular polygon all meet in a single point,
the center of the polygon.
Symmetries of a regular n-gon form the dihedral group Dn of degree n
composed of rotations around the center (by multiples of 2π/n) & re�ec-
tions across lines passing through the center and some vertex.
3
σ_1
C
1
σ_2
σ_3
2
3 EXAMPLES
Since there are n di�erent re�ection axes (the medians for odd n, and for
even n the medians together with the diagonals passing through opposite
vertices) and n di�erent rotations, the order of the dihedral group is∣∣Dn∣∣=2n.Group structure can be described (for �nite groups) using the Cayley
(multiplication) table.
1G . . . h . . .
1G 1G . . . h . . .
......
. . ....
. . .
g g . . . gh . . .
......
. . ....
. . .
3 EXAMPLES
Symmetries map the polygon onto itself any set of distinguished sub-
�gures (like vertices, edges, medians, etc.) is also mapped onto itself.
Consequence: to each symmetry transformation corresponds a per-
mutation of the set X of distinguished sub�gures, hence there exists a
homomorphism πX : Dn → Sym(X) into the symmetric group over X
(can be handy for the computation of the multiplication table).
Since the set V of vertices of a regular n-gon has cardinality n, the
homomorphism πV is bijective only for n=3, when
∣∣Sym(V )∣∣=n!=2n= ∣∣Dn∣∣
3 EXAMPLES
1 C C2 σ1 σ2 σ3 πV
1 1 C C2 σ1 σ2 σ3 ()
C C C2 1 σ3 σ1 σ2 (1, 2, 3)
C2 C2 1 C σ2 σ3 σ1 (1, 3, 2)
σ1 σ1 σ2 σ3 1 C C2 (2, 3)
σ2 σ2 σ3 σ1 C2 1 C (1, 3)
σ3 σ3 σ1 σ2 C C2 1 (1, 2)
Multiplication table of D3 and the homomorphism πV : D3→S3.
3 EXAMPLES
Platonic solid (regular polyhedron): convex spatial �gure all of whose
bounding facets are congruent regular polygons.
Five Platonic solids (up to similarity):
1. tetrahedron (4 triangles);
2. octahedron (8 triangles);
3. icosahedron (20 triangles);
4. cube (6 squares);
5. dodecahedron (12 pentagons).
3 EXAMPLES
3 EXAMPLES
Symmetry groups of regular polyhedra:
· T ∼= A4 (tetrahedron),
· O ∼= S4 (octahedron cube)
· I ∼= A5 (icosahedron dodecahedron).
Regular convex polytopes in 4D: simplex (5-cell), orthoplex (16-cell),
hypercube (8-cell), 600-cell, 120-cell, 24-cell.
Only 3 regular polytopes in dimensions >4: the simplex, the orthoplex
(cross-polytope) and the hypercube.
3 EXAMPLES
3.5 Molecular symmetry groups
Charge density in molecules invariant under a �nite group (point group)
of geometric transformations, leading to restrictions on
1. the structure of the molecular spectrum (Wigner, Tisza);
2. electromagnetic characteristics (dipole and magnetic moments;
3. chemical properties.
3 EXAMPLES
Point groups (in 3D): 7 in�nite families + 7 polyhedral groups
1. Cn∼=Zn (cyclic), e.g. hydrogen peroxide H2O2 (n=2);
2. Cnv∼=Dn (pyramidal), e.g. water (n=2), ammonia (n=3), hydro-
gen �uoride (n=∞);
3. Cnh∼=Zn×Z2 , e.g. boric acid (n=3);
4. Dn∼=Dn (dihedral), e.g. twistane C10H16 (n=2);
5. Dnv ∼= D2n (anti-prismatic), e.g. ethane C2H6 (n = 3), ferrocene
Fe (C5H5)2 (n=5);
6. Dnh∼=Dn×Z2 (prismatic), e.g. ethylene C2H4 (n=2), boron tri�u-
oride BF3 (n=3), fullerene C70 (n=5), carbon dioxide CO2 (n=∞);
3 EXAMPLES
7. Sn∼=Z2n;
8. T∼=A4 (chiral tetrahedral);
9. Td ∼=S4 (tetrahedral), e.g. methane CH4;
10. Th∼=A4×Z2 (pyritohedral);
11. O∼=S4 (chiral octahedral);
12. Oh∼=S4×Z2 (octahedral), e.g. sulfur hexa�uoride SF6;
13. I∼=A5 (chiral icosahedral);
14. Ih∼=A5×Z2 (icosahedral), e.g. C60 fullerene.
3 EXAMPLES
3.6 Crystalline symmetry groups
Some homogeneous substances exhibit anisotropic behavior on a macro-
scopic scale (e.g. mechanical, optical, electric, etc. properties of crys-
tals).
At the phenomenological level: material characteristics like permittivity,
conductivity, etc. are tensorial (rather than scalar) quantities.
Since both microscopic homogeneity, both unordered microscopic inho-
mogeneity leads to isotropic behavior, anisotropy is a consequence of
ordered microscopic inhomogeneity.
Macroscopic order from discrete translational symmetries.
3 EXAMPLES
Crystalline substance: microscopic components (atoms, molecules or
ions) distributed periodically in space.
Quasi-crystal: aperiodic structure exhibiting long range order.
3 EXAMPLES
In a crystal, the microscopic components are localized (in the absence
of defects) around the lattice points of a 3D periodic lattice (the crystal
lattice).
Space group: transformations mapping the crystal lattice into itself
(composed of translations, rotations, re�ections, inversions and various
combinations of the above).
3 EXAMPLES
Translation subgroup: group of pure translations taking the crystal lat-
tice into itself.
Point group: �nite group describing the rotational and re�exion symme-
tries of a crystal.
Crystal structures grouped into crystal classes, families and systems ac-
cording to their point groups and translation subgroups.
3 EXAMPLES
Order of transformation: smallest positive integer N such that the Nth
power of the transformation is the identity.
Crystallographic restriction: the number of integers coprime to the order
of any element of the point group cannot exceed the dimension (valid only
for periodic structures, not quasi-crystals).
dimension 2, 3 4, 5 6, 7
allowed values {2, 3, 4, 6} ∪{5, 8, 10, 12} ∪{7, 9, 14, 18}
Consequence: only �nitely many di�erent crystal structures in any di-
mension.
3 EXAMPLES
Classi�cation results
dimension 2 3† 4‡ 5∗ 6∗
# crystal systems 4 7 33 59 251
# point groups 10 32 227 955 7104
# space groups 17 230 4894 222097 28934974
†: Fedorov (1891), Schön�ies (1891) and Barlow (1894).
‡: Brown, Bülow and Neubüser (1978).
∗: Plesken and Schulz (2000).
3 EXAMPLES
3.7 Space-time symmetry groups
Kinematics: description of the movement of material bodies, i.e. of the
time evolution of their mutual emplacements (distances).
Frame of reference: system of bodies with known relative motions.
The relative motions of all bodies of the Universe are completely deter-
mined by their motions with respect to a speci�c frame.
Question: which frames are the most useful?
Answer: inertial frames, in which the inertial motion of isolated bodies
(not interacting with the rest of the Universe) is uniform translation
( inertial frames move at constant speed relative to each other).
3 EXAMPLES
Galileo's relativity principle: all mechanical laws of motion are the same
in all inertial frames.
Newtonian mechanics: action at a distance (forces act instantly, without
any delay) & physical space is 3D Euclidean and time is a 1D continuum.
Basic symmetries of classical mechanics (in inertial frames):
· both space and time are homogeneous (no preferred origin);
· space is isotropic (no preferred directions);
· any reference frame obtained via a boost (constant speed uniform
translation) is again inertial.
3 EXAMPLES
Galilei group G consisting of
· space translations (3 parameters)
· time translations (1 parameter)
· spatial rotations (3 parameters)
· (Galilean) boosts (3 parameters)
· discrete re�ection symmetries
Noether's theorem: to each one-parameter group of continuous
symmetries of a physical system corresponds a conserved quantity.
3 EXAMPLES
Galilean symmetries correspond to universal �rst integrals.
space translations (linear) momentum
spatial rotations angular momentum
time translations energy
(Galilean) boosts center of mass
Poincaré: the symmetry group of Maxwell's equations (the so-called
Poincaré group P) is the isometry group of 4D (homogeneous and isotropic)
Minkowski space, not the Galilei group.
The Galilei group may be obtained by a limiting procedure (Wigner-
Inönü contraction) from the Poincaré group.
3 EXAMPLES
Einstein: Galileo's relativity principle holds for all physical laws (includ-
ing those of electrodynamics), and there is a limit speed (the speed of
light c) for the propagation of physical causes (no action at a distance)
the true symmetry group of physical laws (at arbitrary speeds,
but neglecting the e�ects of gravity) is the Poincaré group P.
The Poincaré group P consists of
· space-time translations (4 parameters)
· space-time rotations, including the spatial rotations and the Lorentz-
boosts (6 parameters)
· 4D re�ections
3 EXAMPLES
Two relativistic �rst integrals: a 4D vector (four-momentum) and a 4D
antisymmetric tensor (angular momentum).
Pauli-Lubanski (pseudo)vector: four-vector combining 4D linear and
angular momentum, describing helicity (spin) states of moving particles.
General relativity: �at Minkowski space-time has to be replaced by a
suitable curved manifold (a solution of Einstein's equations) Poincaré
group has to be replaced by the corresponding isometry group (e.g. the
de Sitter group).
Historical highlightsFundamental conceptsExamples
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