What Is a Group?

Learning Objectives

In Chapter 1 we learned to see the symmetries of a crystal — the rotations, reflections, inversions, and improper rotations that map a lattice onto itself. In this chapter we learn to compute with them. Symmetry stops being a picture and becomes algebra. The first stepping stone is the most important word in the chapter: group.

By the end of this section you will be able to:

1. State the four group axioms — closure, associativity, identity, and inverses — and decide, by inspection of a small set, whether they hold.
2. Recognise two prototype groups that we will reuse for the rest of the chapter: the integers under addition (an infinite, abelian group) and $D_3$, the symmetries of an equilateral triangle (a finite, non-abelian group of order 6).
3. Build and read a Cayley (multiplication) table, and use the Latin-square test to verify the group axioms at a glance.
4. Compute the order of a group, decide whether it is abelian, and identify all of its subgroups.
5. Connect the abstract definition to a real VASP / spglib output line: see exactly where the order $|G|$ appears, why improper operations are flagged with $\det = -1$, and why this number sets a hard ceiling on the speed-up VASP can extract from your calculation.

The big idea: a crystal's symmetries are not just a list — they are a structured list. Composing two symmetries always gives a third symmetry of the same crystal. That “always” is the defining feature of a group, and it is the reason every theorem in this chapter has teeth.

From Symmetry to Algebra

Imagine you blink and someone rotates a crystal by $120^\circ$ about a 3-fold axis. You open your eyes — and you cannot tell that anything has happened. The atoms are exactly where they were. Now the prankster does it again. Still indistinguishable. A third rotation brings the crystal back to its starting orientation. We have just observed three different operations on the crystal — a rotation by 120°, by 240°, and by 360° — and every one of them produced the same physical configuration.

Two questions arise immediately, and they are the questions that gave birth to group theory:

1. Composition. If I do $R$ first and then $S$, I have done some new operation $SR$. Is this new thing also a symmetry of the crystal? (Spoiler: yes, always.)
2. Reversal. Every symmetry should be undoable — there should be another symmetry that puts everything back. Is that true? (Spoiler: yes, always.)

The minute you write down those two demands carefully, you are no longer doing geometry. You are doing algebra. The set of symmetries of an object — together with the rule for composing two of them — has exactly the structure that nineteenth-century mathematicians (Galois, Cauchy, Cayley) had isolated and called a group. The pay-off is enormous: every theorem about abstract groups is automatically a theorem about symmetries of crystals. The selection rules in spectroscopy, the systematic absences in diffraction, the irreducible representations that label your DFT band states — all of it falls out of one tiny set of axioms.

The Four Axioms — Definition of a Group

A group is a pair $(G, \,\cdot\,)$ consisting of a set $G$ and a binary operation $\cdot \, : G \times G \rightarrow G$ (read: “the product of any two elements of $G$ is again an element of $G$”) satisfying four axioms. The operation $\cdot$ is usually written as juxtaposition, so we write $gh$ instead of $g \cdot h$.

That is the entire definition. Four lines. Memorising the lines is easy; understanding why each one is non-negotiable takes a moment of staring at concrete examples — which is what we do next.

Axiom 1 — Closure

Closure says: the operation can't leak out of the set. If you take any two members of $G$, multiply them, the result must still be inside $G$. For symmetry operations this is almost a tautology: if rotating a crystal by $120^\circ$ leaves it invariant, and reflecting it across a mirror plane also leaves it invariant, then doing the rotation and then the reflection still leaves it invariant. Therefore the composition is again a symmetry. The set is closed.

Axiom 2 — Associativity

Associativity says: parentheses don't matter. When you compose $A$, $B$, and $C$ in that order, you get the same result whether you first do $AB$ and then apply $C$, or first do $BC$ and then apply that to whatever $A$ is acting on. Geometrically: doing three transformations in a fixed order is one well-defined motion of the universe — re-grouping which two you mentally bundle together can't change the final position of an atom.

For matrices this is automatic: matrix multiplication is associative because $((AB)C)_{ik} = \sum_j (AB)_{ij} C_{jk} = \sum_{j,\ell} A_{i\ell} B_{\ell j} C_{jk} = (A(BC))_{ik}$. Since every crystallographic symmetry is a matrix, associativity is free.

Axiom 3 — Identity

The identity says: doing nothing is a thing. There must be a special element $e \in G$ that, when composed with any $g$, returns $g$. For symmetries the identity is the “leave every atom where it is” operation — physically trivial, mathematically essential. Without it we could never write $g \cdot g^{-1} = e$, so we could never express the idea of an inverse.

A nice consequence: the identity is unique. If $e$ and $e'$ were both identities, then $e = e \cdot e' = e'$. So we can speak of the identity, never an identity.

Axiom 4 — Inverses

Every symmetry is reversible. If you rotate a crystal by $+120^\circ$, the rotation by $-120^\circ$ (equivalently $+240^\circ$) is also a symmetry, and the two compose to the identity. If you reflect across a mirror plane, doing it again undoes it. In matrix language this is the statement that every symmetry matrix is invertible — and indeed every orthogonal matrix is, with inverse equal to its transpose.

Example 1 — Integers Under Addition

Before we charge into crystal symmetries, let us verify the axioms in a setting every student already knows. Take $G = \mathbb{Z}$ (the integers $\ldots, -2, -1, 0, 1, 2, \ldots$) and let the operation be ordinary addition.

All four axioms hold, so $(\mathbb{Z}, +)$ is a group. It also happens to satisfy $a + b = b + a$, so it is abelian — but commutativity is not part of the definition. It is a bonus property that some groups have and others (like the triangle group below) emphatically do not. $(\mathbb{Z}, +)$ is also infinite: you can keep adding 1 forever. A crystal's point group is by contrast always finite — but the lattice translation group is infinite, and $(\mathbb{Z}, +)$ is its 1D model.

Example 2 — Symmetries of an Equilateral Triangle

Now the example you came here for. Take an equilateral triangle in a plane, with vertices labelled $1$, $2$, $3$. Which rigid motions map the triangle to itself, paying no attention to the labels? There are exactly six:

This six-element set is the dihedral group $D_3$ (also written $D_{3h}$ when we sit it in 3D and add an horizontal mirror). It is the smallest non-abelian group: $r \cdot s_1 \neq s_1 \cdot r$, as you can verify in the lab below. And although it has only six elements, every feature you will meet later in the chapter — irreducible representations, character tables, selection rules — already shows up here in miniature.

Each element is a $2 \times 2$ orthogonal matrix in the plane. With vertex 1 placed at the top of the triangle, the matrices are:

$r = \begin{pmatrix} -\tfrac{1}{2} & -\tfrac{\sqrt{3}}{2} \\[3pt] \tfrac{\sqrt{3}}{2} & -\tfrac{1}{2} \end{pmatrix}$, $r^2 = \begin{pmatrix} -\tfrac{1}{2} & \tfrac{\sqrt{3}}{2} \\[3pt] -\tfrac{\sqrt{3}}{2} & -\tfrac{1}{2} \end{pmatrix}$, $s_1 = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$, $s_2 = \begin{pmatrix} \tfrac{1}{2} & \tfrac{\sqrt{3}}{2} \\[3pt] \tfrac{\sqrt{3}}{2} & -\tfrac{1}{2} \end{pmatrix}$, $s_3 = \begin{pmatrix} \tfrac{1}{2} & -\tfrac{\sqrt{3}}{2} \\[3pt] -\tfrac{\sqrt{3}}{2} & -\tfrac{1}{2} \end{pmatrix}$.

Notice the determinants: $\det r = \det r^2 = +1$ (proper rotations) and $\det s_1 = \det s_2 = \det s_3 = -1$ (improper). This proper/improper split is the same one we met in Chapter 1 — and it is exactly the $\det = \pm 1$ column you will see in the spglib output at the end of this section.

Interactive D₃ Lab

The fastest way to build genuine intuition for a group is to play with one. Click the buttons below to compose D₃ operations in any order. The triangle morphs from its starting position to the final state. Watch three things happen simultaneously:

• The click sequence at the top right grows — every click adds one factor to the composition.
• The whole sequence collapses, in real time, to a single element of D₃ (shown in green). This is closure: no matter how many you stack, you never escape the six-element set.
• The Cayley table at the bottom highlights, in green, the cell that justifies the latest binary step. After many clicks you will have lit up enough cells to convince yourself the whole table is correct.

A few experiments worth running before you read on:

1. Click $r$ three times. The sequence reduces to $e$: a 360° rotation is the identity. Equivalently, $r^3 = e$ — the order of $r$ as an element is 3.
2. Click any reflection twice. The triangle returns to its starting position. So $s_k^2 = e$ for every $k$: every reflection is its own inverse.
3. Click $r$ then $s_1$. The result is $s_3$. Now reset and click $s_1$ then $r$. The result is $s_2$. Different. This is the proof, in your hands, that $D_3$ is non-abelian.
4. Click any operation followed by its inverse (for reflections, click it twice; for $r$, click $r$ then $r^2$). You always land on $e$. That is axiom G4 in motion.

Reading the Cayley Table

The 6×6 grid in the lab is the Cayley table (or multiplication table) of $D_3$. The convention is: the entry at row $g$, column $h$ is the product $g \cdot h$, meaning “apply $h$ first, then $g$.”

Two visual properties of the table tell you instantly that $D_3$ is a group:

• The identity row and column repeat the headers. The $e$ row reads exactly $e, r, r^2, s_1, s_2, s_3$. That is axiom G3.
• Every element appears exactly once in every row, and exactly once in every column. That is the Latin-square property, and it is equivalent to the existence of inverses (G4) plus closure (G1). Whenever you build a multiplication table of a finite group, this pattern must emerge — if it doesn't, you have made an arithmetic error or your set is not a group.

The table also reveals non-commutativity at a glance: it is not symmetric across its main diagonal. The entry at $(r, s_1)$ is $s_3$, but the entry at $(s_1, r)$ is $s_2$. Reflect the table across its diagonal and it changes — which means the group operation is not commutative.

Order, Abelian, and Subgroups

Three pieces of basic vocabulary will recur in every chapter and every VASP run. Memorise them now.

Order of a group

The order of a finite group $G$, written $|G|$, is simply the number of elements. For $D_3$, $|D_3| = 6$. For the cubic point group $O_h$ (which we will meet later) it is $|O_h| = 48$. For the integers, $|\mathbb{Z}| = \infty$. We also speak of the order of an element: the smallest positive integer $n$ such that $g^n = e$. In $D_3$, the order of $r$ is 3 and the order of every $s_k$ is 2.

Abelian groups

A group is abelian (after Niels Abel) if $gh = hg$ for every pair $g, h \in G$. The integers under addition are abelian; $D_3$ is not. Every cyclic group — a group generated by a single element, like $\{e, r, r^2\}$ inside $D_3$ — is abelian. Whether your crystal's point group is abelian decides things as practical as whether all of its irreducible representations are one-dimensional (abelian → yes; non-abelian → there will be 2D or 3D irreps).

Subgroups

A subgroup $H \subseteq G$ is a subset that is itself a group under the same operation. $D_3$ has six subgroups:

Notice every subgroup's order divides $|D_3| = 6$. That is no coincidence — it is Lagrange's theorem: the order of any subgroup divides the order of the group. Lagrange will reappear when we count how many irreducible representations a group has and when we figure out how many k-points are equivalent in the Brillouin zone.

Why This Matters for Crystals

Everything in the rest of the chapter is built from the four axioms. Three direct pay-offs are worth previewing now:

1. Point groups (next sections). Every crystal's set of point symmetries is a group — one of the 32 crystallographic point groups. The four axioms guarantee that a crystal cannot have a “rogue” symmetry that fails to compose with the others; the symmetry list is structurally complete.
2. Representations and character tables. Every group acts on a vector space; the way it acts is a representation. Decomposing those representations into irreducible pieces gives us the labels we use for orbitals, phonons, and band states. Without group axioms, decomposition makes no sense.
3. Selection rules. A matrix element $\langle \psi_i | \hat{O} | \psi_j \rangle$ is forced to zero whenever the product of the three irreducible labels does not contain the trivial representation. That single rule explains every “forbidden transition” in optical, IR, and Raman spectra. It is a one-page consequence of the group axioms plus orthogonality.

The most-used theorem in this book. If $G$ is the symmetry group of a crystal and $|G| = N$, then in any Brillouin-zone integral the number of k-points VASP actually has to compute is at most $N_\text{full} / N$. The factor $N$ is the order of the group. Mastering this single section is what turns “$120{,}000$ k-points” into “$2{,}500$ k-points” on your screen.

VASP Connection — Group Order in spglib

Time to make the abstract definition concrete. VASP's symmetry analysis (and the OUTCAR line “Found N space group operations”) is powered under the hood by spglib. We will reproduce its analysis from Python on a tiny structure designed to have exactly $D_3$ symmetry — three identical atoms at the vertices of an equilateral triangle inside a hexagonal cell.

Click any line of the code below to see what each variable holds, exactly which arguments each library function consumes, and the output value at that line. Note especially the loop trace: every one of the six rotation matrices spglib returns is shown explicitly, and you can match each one to an element of $D_3$ from the lab above.

      x      y     z
0   1/3   1/3   0.5
1   2/3   1/3   0.5
2   1/3   2/3   0.5

Order of the symmetry group: |G| = 6
  op  0: det = +1  (proper)
  op  1: det = +1  (proper)
  op  2: det = +1  (proper)
  op  3: det = -1  (improper)
  op  4: det = -1  (improper)
  op  5: det = -1  (improper)

1import numpy as np
2import spglib
3
4# Equilateral triangle of identical atoms in a hexagonal cell.
5# This 2D-like motif is the cleanest place to *see* a group of order 6.
6a = 3.0
7lattice = np.array([
8    [a, 0.0, 0.0],
9    [-a / 2, a * np.sqrt(3) / 2, 0.0],
10    [0.0, 0.0, 10.0],   # large c to decouple layers
11])
12
13# Three identical atoms placed at the vertices of an equilateral triangle
14# inside the unit cell, centred at the origin.
15positions = np.array([
16    [1 / 3, 1 / 3, 0.5],   # vertex 1
17    [2 / 3, 1 / 3, 0.5],   # vertex 2
18    [1 / 3, 2 / 3, 0.5],   # vertex 3
19])
20numbers = [1, 1, 1]   # all the same species
21
22cell = (lattice, positions, numbers)
23
24# Ask spglib for the symmetry group.
25sym = spglib.get_symmetry(cell, symprec=1e-5)
26n_ops = len(sym["rotations"])
27print(f"Order of the symmetry group: |G| = {n_ops}")
28
29# List every operation as a 3x3 rotation matrix in fractional coordinates.
30for k, R in enumerate(sym["rotations"]):
31    det = int(round(np.linalg.det(R)))
32    label = "proper" if det == +1 else "improper"
33    print(f"  op {k:2d}: det = {det:+d}  ({label})")

The expected output:

Order of the symmetry group: |G| = 6
  op  0: det = +1  (proper)
  op  1: det = +1  (proper)
  op  2: det = +1  (proper)
  op  3: det = -1  (improper)
  op  4: det = -1  (improper)
  op  5: det = -1  (improper)

Six operations. Three with $\det = +1$ (the rotations $e$, $r$, $r^2$) and three with $\det = -1$ (the reflections $s_1$, $s_2$, $s_3$). This is exactly what the four axioms predict — and exactly what a VASP run on this structure would print near the top of its OUTCAR.

Summary

1. A group is a set $G$ with a binary operation that satisfies four axioms: closure, associativity, identity, and inverses.
2. The integers under addition $(\mathbb{Z}, +)$ are an infinite abelian group, the model for crystal translations.
3. The dihedral group $D_3$, with elements $\{e, r, r^2, s_1, s_2, s_3\}$, is the smallest non-abelian group and the prototype for the rest of the chapter.
4. The Cayley table of any group is a Latin square. Symmetry across its main diagonal characterises abelian groups.
5. The order $|G|$ is the number of elements. Every subgroup's order divides $|G|$ (Lagrange).
6. spglib (and through it, VASP) computes the symmetry group of a crystal automatically; reading its output is just reading the group's elements as 3×3 integer matrices in the fractional basis.

Looking ahead. In the next section we put groups to work on the 32 crystallographic point groups themselves — naming them, listing their elements, and building the multiplication table for each. The four axioms you just memorised will quietly do all the heavy lifting.