Groups

A group is a set of operations that can be composed, undone, and applied in a consistent order. In physics, groups are the algebraic language of symmetry: translations compose with translations, rotations compose with rotations, phase changes compose with phase changes, and permutations compose with permutations.

Quantum mechanics uses groups in two closely related ways. First, a group may describe transformations of a physical system. Second, a group may be represented by operators on a Hilbert space. This page owns the basic group concept. Group Actions explains what the transformations act on; Representations explains how group elements become linear operators; Lie Groups adds smooth continuous parameters; Lie Algebras extracts the infinitesimal commutator structure.

Definition

A group is a set $G$ with a binary operation

G\times G\to G, \qquad (g,h)\mapsto gh,

satisfying four axioms.

Closure:

g,h\in G \quad\Longrightarrow\quad gh\in G.

Associativity:

(gh)k = g(hk)

for all $g,h,k\in G$ .

Identity:

\exists e\in G \quad \text{such that} \quad eg=ge=g

for all $g\in G$ .

Inverses:

\forall g\in G, \quad \exists g^{-1}\in G \quad \text{such that} \quad g^{-1}g=gg^{-1}=e.

The operation is often called multiplication even when it is not ordinary numerical multiplication. For additive groups, one writes $g+h$ , the identity as $0$ , and the inverse as $-g$ .

Symmetry as Composition

A symmetry transformation is useful only if transformations can be composed. If one rotates a system and then rotates it again, the result should be another allowed rotation. If a transformation can be physically undone, its inverse should also be allowed.

This is exactly what the group axioms formalize. A group element is not necessarily a number. It may be:

a rotation of space;
a translation;
a phase multiplication;
a permutation of identical labels;
a matrix acting on vectors;
an abstract symbol whose multiplication rules encode a symmetry.

The identity element represents doing nothing. The inverse element represents undoing a transformation. Associativity says that a chain of transformations has an unambiguous result once their order is fixed.

Order Matters

A group is abelian if

gh=hg

for every $g,h\in G$ . Otherwise it is nonabelian.

Many simple phase and translation groups are abelian. Three-dimensional rotations are not. If $R_x$ is a rotation about the $x$ axis and $R_y$ is a rotation about the $y$ axis, then generally

R_xR_y\ne R_yR_x.

This noncommutativity is the finite-transformation shadow of the angular-momentum commutators in quantum mechanics.

Basic Examples

The integers $\mathbb Z$ form a group under addition. The identity is $0$ , and the inverse of $n$ is $-n$ .

The residue classes modulo $N$ form the cyclic group $\mathbb Z_N$ under addition modulo $N$ . This is the simplest finite group family. For $N=2$ , it describes a yes-or-no operation such as parity labels or a sign flip.

The nonzero complex numbers $\mathbb C^\times$ form a group under multiplication. The identity is $1$ , and the inverse of $z$ is $z^{-1}$ .

The unit complex numbers form the circle group

U(1) = \{e^{i\theta}:\theta\in\mathbb R\}.

This group describes phases. Its operation is multiplication:

e^{i\theta}e^{i\phi} = e^{i(\theta+\phi)}.

The invertible $n\times n$ complex matrices form the general linear group $GL(n,\mathbb C)$ under matrix multiplication. The unitary matrices form $U(n)$ , and the determinant-one unitary matrices form $SU(n)$ .

The permutations of $n$ objects form the symmetric group $S_n$ . This group is central for identical particles because exchanging labels must leave physical predictions unchanged, even though bosonic and fermionic states respond differently to the exchange.

Matrix Groups

Many quantum-mechanical groups are matrix groups: their elements are matrices and their operation is matrix multiplication.

Examples include:

$U(n)$ , the group of $n\times n$ unitary matrices;
$SU(n)$ , the subgroup of $U(n)$ with determinant one;
$SO(3)$ , the group of real three-dimensional rotations;
$SU(2)$ , the group central to spin- $1/2$ and two-level systems.

Unitary matrix groups are especially important because unitary transformations preserve Hilbert-space inner products. In quantum mechanics, this is tied to preservation of transition probabilities.

Subgroups

A subgroup $H$ of a group $G$ is a subset that is itself a group under the same operation. One writes

H\le G.

For example, $SU(n)$ is a subgroup of $U(n)$ :

SU(n)\le U(n).

The subgroup relation is a way of saying that a smaller set of transformations closes among itself. Physically, a Hamiltonian may have a full symmetry group, or only a subgroup after perturbations or external fields are added.

Cyclic Groups and Generators

A group is cyclic if one element generates every element by repeated application. If $a$ is a generator, the group consists of powers

\ldots,a^{-2},a^{-1},e,a,a^2,\ldots.

For a finite cyclic group of order $N$ ,

a^N=e.

In quantum mechanics, cyclic groups appear in discrete rotations, lattice translations with periodic boundary conditions, phase factors at roots of unity, and parity-like two-element symmetries.

The word “generator” has a broader meaning for continuous groups, where Hermitian operators generate one-parameter unitary families. That infinitesimal notion belongs to Lie groups and Lie algebras; the shared intuition is that a small amount of data can produce many group elements.

Homomorphisms and Isomorphisms

A homomorphism from $G$ to $K$ is a map

\varphi:G\to K

that preserves multiplication:

\varphi(gh) = \varphi(g)\varphi(h).

An isomorphism is a bijective homomorphism whose inverse is also a homomorphism. Isomorphic groups have the same abstract multiplication structure, even if their elements are written differently.

This distinction matters in physics. A group may be described abstractly, by matrices, by transformations of a space, or by operators on states. These descriptions can encode the same group law while looking very different.

From Groups to Quantum Symmetries

In quantum mechanics, a symmetry group is usually realized by transformations of rays, states, or observables. In a unitary representation, one assigns an operator $U(g)$ to each group element $g$ so that

U(gh) = U(g)U(h).

This equation says that the operator assigned to a composed symmetry is the composed operator. Because operator multiplication is order-sensitive, the convention for whether $gh$ means “do $h$ first, then $g$ ” must match the convention used for transformations.

The representation may be ordinary, projective, unitary, antiunitary, finite-dimensional, or infinite-dimensional depending on the physical problem. The basic group page does not classify those cases. It supplies the multiplication language that those later distinctions use.

Physical Examples

Spatial translations form a group. On the line, translations by real distances add:

T(a)T(b) = T(a+b).

In quantum mechanics, translations are represented by unitary operators generated by momentum.

Rotations in three-dimensional space form $SO(3)$ . Quantum angular momentum arises because rotations of states are represented on Hilbert space, and the corresponding infinitesimal generators satisfy noncommuting algebraic relations.

Phase transformations form $U(1)$ . Global phase multiplication of one state vector is physically redundant, while a genuine $U(1)$ symmetry of a system can imply a conserved quantity through its generator.

Permutations form $S_n$ . Identical-particle quantum mechanics uses representations of permutation groups to organize bosonic, fermionic, and more general exchange structure.

The group $SU(2)$ is not the same as $SO(3)$ , but it is closely related. Its double-cover relation to rotations explains why spin- $1/2$ states can change sign under a $2\pi$ rotation.

What the Group Does Not Yet Tell You

Knowing the abstract group is not the same as knowing how it acts.

The group $G$ tells you which transformations can be composed. A group action tells you what the transformations act on. A representation tells you how group elements become linear or antilinear operators on a vector space. A Lie algebra tells you the infinitesimal structure near the identity of a continuous group.

For example, the same group $\mathbb Z_2$ can act as spatial parity, spin flip, particle–hole transformation, or a sign change of a field. The abstract multiplication rule is the same, but the physical meaning depends on the action or representation.

Common Mistakes

Calling a set a group without checking inverses and closure.
Forgetting that matrix multiplication and rotations are generally noncommutative.
Confusing an abstract group with a particular matrix representation of that group.
Treating a symmetry group as a symmetry of a Hamiltonian before checking that the Hamiltonian is invariant.
Assuming that a group element must be a number rather than a transformation.
Confusing the finite-generator idea in cyclic groups with infinitesimal generators of Lie groups.
Ignoring composition order when translating between transformations and operators.

Cross-Links

References

M. Artin, Algebra, 2nd ed., Pearson, 2011.
D. S. Dummit and R. M. Foote, Abstract Algebra, 3rd ed., Wiley, 2004.
B. C. Hall, Lie Groups, Lie Algebras, and Representations, 2nd ed., Springer, 2015.
E. P. Wigner, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, 1959.
J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
H. Georgi, Lie Algebras in Particle Physics, 2nd ed., Westview Press, 1999.

Exercises

Show that $\mathbb Z_N$ under addition modulo $N$ is a group.

Solution

Closure holds because adding two residue classes modulo $N$ gives another residue class modulo $N$ . Associativity comes from associativity of integer addition. The identity is the class of $0$ . The inverse of the class of $k$ is the class of $N-k$ , with the class of $0$ as its own inverse.

Let $P$ be a transformation with $P^2=I$ . Show that $\{I,P\}$ is a group under composition.

Solution

Closure follows from $II=I$ , $IP=PI=P$ , and $PP=P^2=I$ . Composition is associative because function composition or matrix multiplication is associative. The identity is $I$ . The inverse of $I$ is $I$ , and the inverse of $P$ is $P$ itself.

Explain why the real numbers $\mathbb R$ do not form a group under multiplication, while the nonzero real numbers $\mathbb R^\times$ do.

Solution

The number $0$ has no multiplicative inverse, so $\mathbb R$ fails the inverse axiom under multiplication. Removing zero gives $\mathbb R^\times$ . The product of two nonzero real numbers is nonzero, multiplication is associative, the identity is $1$ , and every nonzero real number $x$ has inverse $1/x$ .

Suppose a map $\varphi:G\to K$ satisfies $\varphi(gh)=\varphi(g)\varphi(h)$ . Show that if $e_G$ is the identity in $G$ , then $\varphi(e_G)$ acts as an identity on every element in the image of $\varphi$ .

Solution

For any element in the image, write it as $\varphi(g)$ . Since $e_Gg=g$ and $ge_G=g$ ,

\varphi(e_G)\varphi(g) = \varphi(e_Gg) = \varphi(g),

and

\varphi(g)\varphi(e_G) = \varphi(ge_G) = \varphi(g).

Thus $\varphi(e_G)$ is the identity on the image of $\varphi$ .

Why does the abstract group $\mathbb Z_2$ not by itself tell you whether a physical symmetry is parity, spin flip, or particle exchange?

Solution

The abstract group only says that there are two elements and that the nonidentity element squares to the identity. It does not specify what space the elements act on or how they act. Parity, spin flip, and a two-particle exchange can share the same abstract multiplication rule while having different physical actions and different operator representations.