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Section 1.1 Groups

Subsection 1.1.1 Definition

In physics, we think of symmetries in terms of transformations of a system (think of rotations), that we can compose with each other. What we want to do here is capture the abstract properties that are shared by all such symmetry transformations. This is what the abstract concept of a group does. Basically, a group is a set with a binary operation (composition) that satisfies a bunch of axioms.

Definition 1.1.1. Group.

A group is a set \(G\) with a binary operation \(\cdot\text{,}\) which we call composition or multiplication, that satisfies the following axioms:

  • Closure: For all \(a, b \in G\text{,}\) \(a \cdot b \in G\text{.}\)
  • Associativity: For all \(a,b,c \in G\text{,}\) \((a \cdot b) \cdot c = a \cdot (b \cdot c)\text{.}\)
  • Identity: There exists a group element \(e \in G\text{,}\) known as the identity, such that \(e \cdot a = a \cdot e = a\) for all \(a \in G\text{.}\)
  • Inverses: For all \(a \in G\text{,}\) there exists an element \(a^{-1} \in G\) such that \(a \cdot a^{-1} = a^{-1} \cdot a = e\text{.}\)

It is easy to show a few elementary properties of groups:

  1. The identity element of a group is always unique.
  2. For each element \(a \in G\text{,}\) there is a unique inverse \(a^{-1} \in G\text{.}\)
  3. In a group, \(a \cdot b = a \cdot c\) implies \(b = c\text{,}\) and \(b \cdot a = c \cdot a\) implies \(b= c\text{,}\) as expected.

Prove these three properties.

We can further categorize groups according to some of their properties:

Definition 1.1.3. Abelian groups.

If \(a \cdot b = b \cdot a\) for all \(a,b \in G\text{,}\) i.e. composition is commutative, then we say that \(G\) is abelian. We say that it is non-abelian otherwise.

Definition 1.1.4. Finite groups.

If \(G\) has a finite number of elements, we say that the group is finite. We call \(|G|\) the order of \(G\text{.}\)

Definition 1.1.5. Continuous and discrete groups.

If we write \(G = \{ g_\alpha \}\text{,}\) then the index \(\alpha\) can be either discrete or continuous. Note that a discrete group can well be infinite.

Subsection 1.1.2 Examples

You already know many examples of groups. Let us look at a few of them:

Rotations in two-dimensional space form a group, denoted by \(SO(2)\text{,}\) which is an abelian continuous group. Rotations in three-dimensional space also form a group, which is denoted by \(SO(3)\text{,}\) which happens to be non-abelian. Similarly, rotations in \(n\) dimensions form the non-abelian group \(SO(n)\text{.}\)

The set \(\mathbb{R}\) with the operation of addition forms an abelian continuous group. If we restrict to the subset of integers \(\mathbb{Z}\text{,}\) we still get an infinite abelian group, but it is now discrete.

The group with only one element \(\{1\}\) forms a group under multiplication, but it is a rather boring one. It is called the trivial group.

The two square roots of 1, \(\{1,-1 \}\text{,}\) form the group \(\mathbb{Z}_2\) under multiplication, which is an abelian discrete group of order 2. More generally, the \(N\)'th roots of unity, \(\{ e^{2 \pi i k /N } \}_{k=0,\ldots,N-1}\) form the group \(\mathbb{Z}_N\) under multiplication, which is an abelian discrete group of order \(N\text{.}\)

Complex numbers of norm 1, namely \(e^{i \theta}\) with \(\theta \in [0, 2 \pi)\text{,}\) form an abelian continuous group under multiplication called \(U(1)\text{.}\)

Addition of integers mod \(N\) generates an abelian discrete group of order \(N\text{.}\) The elements of the set are \(\{0,1,\ldots,N-1\}\text{,}\) with the operation of addition mod \(N\text{.}\) This has in fact the same abstract group structure as \(\mathbb{Z}_N\) defined above.

The set of invertible mappings \(f: S \to S\) of a finite set S with n elements forms a group, which is denoted \(S_n\) and called symmetric group (or permutation group). The elements of the group \(S_n\) are permutations of n objects. There are \(n!\) such distinct permutations, so \(S_n\) has order \(n!\text{,}\) and is non-abelian for \(n \geq 3\text{.}\)

The set of invertible linear maps \(f: V \to V\) of a vector space \(V\) forms a continuous group, known as the general linear group of \(V\) and denoted by \(GL(V)\text{.}\) If \(V = \mathbb{R}^n\text{,}\) we write \(GL(n,\mathbb{R})\text{;}\) if \(V = \mathbb{C}^n\text{,}\) we write \(GL(n,\mathbb{C})\text{.}\) We can also think of those as the sets of invertible \(n \times n\) matrices with the operation of matrix multiplication (with either real or complex entries).

The set of \(n \times n\) matrices with unit determinant under matrix multiplication also forms a continuous group, since \(\det (A B) = \det (A) \det (B)\text{.}\) This is called the special linear group \(SL(n, \mathbb{R})\) for matrices with real entries, and \(SL(n, \mathbb{C})\) for matrices with complex entries.

Lorentz transformations in Minkowski space form a continuous group. If we also include translations, we obtain the Poincare group.

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