Examples of groups
Since the definition of a group is so abstract, let us try to examine some examples of groups.
As a first, let us consider a group that is familiar to all of us. Let us take the set \(\mathbb{R}_0\): the rational numbers excluding \(0\), together with the operation of multiplication. We can check that every group axiom holds (the identity element is \(1\), and we know the inverse of every real number), even the commutative one. We can therefore say that \(\mathbb{R}_0\) with multiplication is an Abelian group.
We gave a general name to the group operation: group multiplication. This doesn’t mean that the group operation can’t be addition, for example, as group multiplication is just a name. A perfectly valid example of an Abelian group is the set of integers \(\mathbb{Z}\), together with addition. Again, we can check that all group axioms hold (the identity element is \(0\), and we all know the inverse of integers with respect to addition).
As a slightly more complicated example of a group, we will consider the general linear group over \(\mathbb{R}\) of degree \(n\), denoted by \(\text{GL}(n, \mathbb{R})\). This is the set of all invertible \(n \times n\)-matrices with real entries, with the operation of matrix multiplication. The identity element is \(I_n\): the \(n \times n\)-identity matrix (a diagonal matrix with \(1\) on the diagonal), and since we have specified the set as being the set of invertible matrices, every matrix has an inverse. We should emphasize that \(\text{GL}(n, \mathbb{R})\) is not Abelian, as, in general, matrix multiplication is not commutative. A special case of this group is formed by requiring that the determinant of the invertible \(n \times n\)-matrices is equal to \(1\). We call this set of matrices, together with matrix multiplication, the special linear group \(\text{SL}(n)\).
Many sets of matrices, together with the operation of multiplication form a group. We have for example \(\text{O}(n)\), being the set of \(n \times n\) orthogonal (\(Q^\text{T} Q = Q Q^\text{T} = I_n\)) matrices under matrix multiplication. A special group that is related to \(\text{O}(n)\) is \(\text{SO}(n)\), being the set of orthogonal matrices with determinant equal to \(1\), under matrix multiplication. Furthermore, we also have the group \(\text{U}(n)\), being the set of \(n \times n\) unitary matrices (\(U U^\dagger = U^\dagger U = I_n\)), under the group operation of matrix multiplication. Again, a special variant is \(\text{SU}(n)\), being the set of \(n \times n\) unitary matrices with determinant equal to \(1\), under matrix multiplication.
The previous examples are examples of matrix Lie groups.
As a first more abstract example, let us take a look at the trivial group. It consists of the set \(G = \set{e}\) under group multiplication. We have to specify that \(e\) is the identity element, and consequently its own inverse. With this in mind, we can check that the trivial group is Abelian.
As another abstract example, let’s take the set of elements \[\begin{equation} \require{physics} G = \set{E, C_2, \sigma_v, \sigma_v'} \thinspace , \end{equation}\] with the multiplication table given in Table \(\ref{table:multiplication_table_C2v}\).
| C\(_\text{2v}\) | \(E\) | \(C_2\) | \(\sigma_v\) | \(\sigma_v'\) |
|---|---|---|---|---|
| \(E\) | \(E\) | \(C_2\) | \(\sigma_v\) | \(\sigma_v'\) |
| \(C_2\) | \(C_2\) | \(E\) | \(\sigma_v'\) | \(\sigma_v\) |
| \(\sigma_v\) | \(\sigma_v\) | \(\sigma_v'\) | \(E\) | \(C_2\) |
| \(\sigma_v'\) | \(\sigma_v'\) | \(\sigma_v\) | \(C_2\) | \(E\) |
A multiplication table is read as follows. Take an element from the first column (for example \(E\)), and take an element of the second column (\(C_2\)), and find their product as \(C_2 \cdot E = C_2\) (note that we read group multiplication conventionally from right to left). This set \(G\), together with the multiplication \(\cdot\) specified in the multiplication table, forms a group as all four group axioms are fulfilled. As commutativity is also fulfilled 2, this group is even Abelian.
We can even introduce bigger sets: \[\begin{equation} G = \set{E, C_3, C^2_3, \sigma_v, \sigma_v', \sigma_v''} \thinspace , \end{equation}\]
| C\(_\text{3v}\) | \(E\) | \(C_3\) | \(C^2_3\) | \(\sigma_v\) | \(\sigma_v'\) | \(\sigma_v''\) |
|---|---|---|---|---|---|---|
| \(E\) | \(E\) | \(C_3\) | \(C^2_3\) | \(\sigma_v\) | \(\sigma_v'\) | \(\sigma_v''\) |
| \(C_3\) | \(C_3\) | \(C^2_3\) | \(E\) | \(\sigma_v'\) | \(\sigma_v''\) | \(\sigma_v\) |
| \(C^2_3\) | \(C^2_3\) | \(E\) | \(C_3\) | \(\sigma_v''\) | \(\sigma_v\) | \(\sigma_v'\) |
| \(\sigma_v\) | \(\sigma_v\) | \(\sigma_v''\) | \(\sigma_v'\) | \(E\) | \(C^2_3\) | \(C_3\) |
| \(\sigma_v'\) | \(\sigma_v'\) | \(\sigma_v\) | \(\sigma_v''\) | \(C_3\) | \(E\) | \(C^2_3\) |
| \(\sigma_v''\) | \(\sigma_v''\) | \(\sigma_v'\) | \(\sigma_v\) | \(C^2_3\) | \(C_3\) | \(E\) |
Given the multiplication table in Table \(\ref{table:multiplication_table_C3v}\), we can verify that the set \(G\), together with the group multiplication forms a non-Abelian group.
An easy way to confirm the commutative property, is to verify that the multiplication table is symmetric with respect to its diagonal.↩︎