# LOG#091. Group theory(XI).

Today, we are going to talk about the Lie groups $U(n)$ and $SU(n)$, and their respective Lie algebras, generally denoted by $u(n)$ and $su(n)$ by the physics community. In addition to this, we will see some properties of the orthogonal groups in euclidena “signature” and general quadratic metrics due to their importance in special relativity or its higher dimensional analogues.

Let us remember what kind of groups are $U(n)$ and $U(n)$:

1) The unitary group is defined by:

$U(n)\equiv\left\{ U\in M_{n\times n}(\mathbb{C})/UU^+=U^+U=I\right\}$

2) The special unitary group is defined by:

$SU(n)\equiv\left\{ U\in M_{n\times n}(\mathbb{C})/UU^+=U^+U=I,\det (U)=1\right\}$

The group operation is the usual matrix multiplication. The respective algebras are denoted as we said above by $u(n)$ and $su(n)$. Moreover, if you pick an element $U\in U(n)$, there exists an hermitian (antihermitian if you use the mathematician “approach” to Lie algebras/groups instead the convention used mostly by physicists) $n\times n$ matrix $H$ such that:

$U=\exp (iH)$

Some general properties of unitary and special unitary groups are:

1) $U(n)$ and $SU(n)$ are compact Lie groups. As a consequence, they have unitary, finite dimensional and irreducible representations. $U(n)$ and $SU(n)$ are subgroups of $U(m)$ if $m\geq n$.

2) Generators or parameters of unitary and special unitary groups. As we have already seen, the unitary group has $n^2$ parameters (its “dimension”) and it has rank $n-1$ (its number of Casimir operators). The special unitary group has $n^2-1$ free parameters (its dimension) and it has rank $n-1$ (its number of Casimir operators).

3) Lie algebra generators. The unitary group has a Lie algebra generated by the space of $n^2$ dimensional complex matrices. The special unitary group has a Lie algebra generated by the $n^2-1$ dimensional space of hermitian $n\times n$ traceless matrices.

4) Lie algebra structures. Given a basis of generators $L_i$ for the Lie algebra, we define the constants $C_{ijk}$, $f_{ijk}$, $d_{ijk}$ by the following equations:

$\left[L_i,L_m\right]=C_{ijk}L_k=if_{ijk}L_k$

$L_iL_j+L_jL_i=\dfrac{1}{3}\delta_{ij}I+d_{ijk}L_k$

These structure constants $f_{ijk}$ are totally antisymmetric under the exchange of any two indices while the coefficients $d_{ijk}$ are symmetric under those changes. Moreover, we also have:

$d_{ijk}=2\mbox{Tr}(\left\{L_i,L_j\right\}L_k)$

$f_{ijk}=-2i\mbox{Tr}(\left[L_i,L_j\right]L_k)$

Remark(I):   From $U=e^{iH}$, we get $\det U=e^{i\mbox{Tr} (H)}$, and from here we can prove the statement 3) above.

Remark(II): An arbitrary element of $U(n)$ can be expressed as a product of an element of $U(1)$ and an element of $SU(n)$. That is, we can write $U(n)\cong U(1)\cup SU(n)$, where the symbol $\cong$ means “group isomorphism”.

Example 1. The SU(2) group.

In particular, for $n=2$, we get

$SU(2)=\left\{U\in M_{2\times 2})(\mathbb{C})/UU^+=U^+U=I_{2\times 2},\det U=1\right\}$

This is an important group in physics! It appears in many contexts: angular momentum (both classical and quantum), the rotation group, spinors, quantum information theory, spin networks and black holes, the Standard Model, and many other places. So it is important to know it at depth. The number of parameters of SU(2) is equal to 3 and its rank is equal to one (1). As generators of the Lie algebra associated to this Lie group, called su(2), we can choose for free 3 any independent traceless (trace equal to zero) matrices. As a convenient choice, it is usual to select the so called Pauli matrices $\sigma_i$:

$\sigma_1=\begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix}$

$\sigma_2=\begin{pmatrix}0 & -i\\ i & 0\end{pmatrix}$

$\sigma_3=\begin{pmatrix} 1 & 0\\ 0 & -1\end{pmatrix}$

In general, these matrices satisfy an important number of mathematical relations. The most important are:

$\left\{\sigma_i,\sigma_j\right\}=2\sigma_i\sigma_j+2i\varepsilon_{ijk}\sigma_k$

and

$\sigma_i\sigma_j=i\varepsilon_{ijk}\sigma_k$

The commutators of Pauli matrices are given by:

$\left[\sigma_i,\sigma_j\right]=2if_{ijk}\sigma_k$

$f_{ijk}=\dfrac{1}{2}\varepsilon_{ijk}$ $d_{ijk}=0$

The Casimir operator/matrix related to the Pauli basis is:

$C(\sigma_i)=\sigma_i^2=\sigma_1^2+\sigma_2^2+\sigma_3^2$

This matrix, by Schur’s lemma, has to be a multiple of the identity matrix (it commutes with each one of the 3 generators of the Pauli algebra, as it can be easily proved). Please, note that using the previous Pauli representation of the Pauli algebra we get:

$\displaystyle{C=\sum_i\sigma_i^2=3I}$

Q.E.D.

A similar relation, with different overall prefactor, must be true for ANY other representation of the Lie group algebra su(2). In fact, it can be proved in Quantum Mechanics that this number is “four times” the $j(j+1)$ quantum number associated to the angular momentum and it characterizes completely the representation. The general theory of the representation of the Lie group SU(2) and its algebra su(2) is known in physics as the general theory of the angular momentum!

Example 2. The SU(3) group.

If n=3, the theory of $SU(3)$ is important for Quantum Chromodynamics (QCD) and the quark theory. It is also useful in Grand Unified Theories (GUTs) and flavor physics.

$SU(3)=\left\{U\in M_{3\times 3})(\mathbb{C})/UU^+=U^+U=I_{3\times 3},\det U=1\right\}$

The number of parameters of SU(3) is 8 (recall that there are 8 different massless gluons in QCD) and the rank of the Lie algebra is equal to two, so there are two Casimir operators.

The analogue generators of SU(3), compared with the Pauli matrices, are the so-called Gell-Mann matrices. They are 8 independent traceless matrices. There are some “different” elections in literature, but a standard choice are the following matrices:

$\lambda_1=\begin{pmatrix}0 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 &0\end{pmatrix}$

$\lambda_2=\begin{pmatrix}0 & -i & 0\\ i & 0 & 0\\ 0 & 0 &0\end{pmatrix}$

$\lambda_3=\begin{pmatrix}1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 &0\end{pmatrix}$

$\lambda_4=\begin{pmatrix}0 & 0 & 1\\ 0 & 0 & 0\\ 1 & 0 &0\end{pmatrix}$

$\lambda_5=\begin{pmatrix}0 & 0 & -i\\ 0 & 0 & 0\\ i & 0 &0\end{pmatrix}$

$\lambda_6=\begin{pmatrix}0 & 0 & 0\\ 0 & 0 & 1\\ 0 & 1 &0\end{pmatrix}$

$\lambda_7=\begin{pmatrix}0 & 0 & 0\\ 0 & 0 & -i\\ 0 & i &0\end{pmatrix}$

$\lambda_8=\dfrac{1}{\sqrt{3}}\begin{pmatrix}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 &-2\end{pmatrix}$

Gell-Mann matrices above satisfy a normalization condition:

$\mbox{Tr}(\lambda_i\lambda_j)=2\delta_{ij}$

where $\delta_{ij}$ is the Kronecker delta in two indices.

The two Casimir operators for Gell-Mann matrices are:

1) $\displaystyle{C_1(\lambda_i)=\sum_{i=1}^8\lambda_i^2}$

This operator is the natural generalization of the previously seen SU(2) Casimir operator.

2) $\displaystyle{C_2(\lambda_i)=\sum_{ijk}d_{ijk}\lambda_i\lambda_j\lambda_k}$

Here, the values of the structure constans $f_{ijk}$ and $d_{ijk}$ for the su(3) Lie algebra can be tabulated in rows as follows:

1) For $ijk=123,147,156,246,257,345,367,458,678$ we have $f_{ijk}=1,\dfrac{1}{2},-\dfrac{1}{2},\dfrac{1}{2},\dfrac{1}{2},\dfrac{1}{2},-\dfrac{1}{2},\dfrac{\sqrt{3}}{2},\dfrac{\sqrt{3}}{2}$.

2) If

$ijk=118,146,157,228,247,256,338,344,355,366,377,448,558,668,778,888$

then have

$d_{ijk}=\dfrac{1}{\sqrt{3}},\dfrac{1}{2},\dfrac{1}{2},\dfrac{1}{\sqrt{3}},-\dfrac{1}{2},\dfrac{1}{2},\dfrac{1}{\sqrt{3}},\dfrac{1}{2},\dfrac{1}{2},-\dfrac{1}{2},-\dfrac{1}{2},-\dfrac{1}{2\sqrt{3}},-\dfrac{1}{2\sqrt{3}},-\dfrac{1}{2\sqrt{3}},-\dfrac{1}{2\sqrt{3}},-\dfrac{1}{\sqrt{3}}$

Example 3. Euclidean groups, orthogonal groups and the Lorentz group in 4D and general $D=s+t$ dimensional analogues.

In our third example, let us remind usual galilean relativity. In a 3D world, physics is the same for every inertial observer (observers moving with constant speed). Moreover, the fundamental invariant of “motion” in 3D space is given by the length:

$L^2=x^2+y^2+z^2=\delta_{ij}x^ix^j$ $\forall i,j=1,2,3$

In fact, with tensor notation, the above “euclidean” space can be generalized to any space dimension. For a ND space, the fundamental invariant reads:

$\displaystyle{L_N^2=\sum_{i=1}^NX_i^2=x_1^2+x_2^2+\cdots+x_N^2}$

Mathematically speaking, the group leaving the above metrics invariant are, respectively, SO(3) and SO(N). They are Lie groups with dimensions $3$ and $N(N-1)/2$, respectively and their Lie algebra generators are antisymmetric traceless $3\times 3$ and $N\times N$ matrices. Those metrics are special cases of quadratic forms and it can easily proved that orthogonal transformations with metric $\delta_{ij}$ (the euclidean metric given by a Kronecker delta tensor) are invariant in the following sense:

$A^\mu_{\;\;\; i}\delta_{\mu\nu}A^\nu_{\;\;\; j}=\delta_{ij}$

or equivalently

$A\delta A^T=\delta$

using matric notation. In special relativity, the (proper) Lorentz group $L$ is composed by every real $4\times 4$ matrix $\Lambda^\mu_{\;\;\;\nu}$ connected to the identity through infinitesimal transformations, and the Lorentz group leaves invariant the Minkovski metric(we use natural units with $c=1$):

$s^2=X^2+Y^2+Z^2-T^2$ if you use the “mostly plus” 3+1 metric ($\eta=\mbox{diag}(1,1,1,-1)$) or, equivalentaly

$s^2=T^2-X^2-Y^2-Z^2$ if with a “mostly minus” 1+3 metric ($\eta=\mbox{diag}(1,-1,-1,-1)$).

These equations can be also genearlized to arbitrary signature. Suppose there are s-spacelike dimensions and t-time-like dimensions ($D=s+t$). The associated non-degenerated quadratic form is:

$\displaystyle{s^2_D=\sum_{i=1}^sX_i^2-\sum_{j=1}^tX_j^2}$

In matrix notation, the orthogonal rotations leaving the above quadratic metrics are said to belong to the group $SO(3,1)$ (or $SO(1,3)$ is you use the mostly minus convention) real orthogonal group over the corresponding quadratic form. The signature of the quadratic form is said to be $S=2$ or $(3,1)$ (equivalently $\Sigma=3-1=2$ and $(1,3)$ with the alternative convention). We are not considering “degenerated” quadratic forms for simplicity of this example. The Lorentzian or Minkovskian metric are invariant in the same sense that the euclidean example before:

$L^\mu_{\;\;\;\alpha}\eta_{\alpha\beta}L^\mu_{\;\;\;\beta}=\eta_{\alpha\beta}$

$LGL^T=G$

The group $SO(s,t)$ has signature $(s,t)$ or $s-t$ or $s+t$ in non-degenerated quadratic spaces. Obviously, the rotation group $SO(3)$ is a subgroup of $SO(3,1)$ and more generally $SO(s)$ is a subgroup of $SO(s,t)$. We are going to focus now a little bit on the common special relativity group $SO(3,1)$. This group have 6 parameters or equivalently its group dimension is 6. The rank of this special relativity group is equal to 1. We can choose as parameters for the $SO(3,1)$ group 3 spatial rotation angles $\omega_i$ and three additional parameters, that we do know as rapidities $\xi_i$. These group generators have Lie algebra generators $S_i$ and $K_i$ or more precisely, if we define the Lorentz boosts as

$\xi=\dfrac{\beta}{\parallel\beta\parallel}\tanh^{-1}\parallel \beta\parallel$

In the case of $SO(3,1)$, a possible basis for the Lie algebra generators are the next set of matrices:

$iS_1=\begin{pmatrix}0 & 0 & 0& 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & -1 & 0\end{pmatrix}$

$iS_2=\begin{pmatrix}0 & 0 & 0& 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\end{pmatrix}$

$iS_3=\begin{pmatrix}0 & 0 & 0& 0\\ 0 & 0 & 1 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0\end{pmatrix}$

$iK_1=\begin{pmatrix}0 & 1 & 0& 0\\ -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\end{pmatrix}$

$iK_2=\begin{pmatrix}0 & 0 & 1& 0\\ 0 & 0 & 0 & 0\\ -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\end{pmatrix}$

$iK_3=\begin{pmatrix}0 & 0 & 0& 1\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ -1 & 0 & 0 & 0\end{pmatrix}$

And the commutation rules for these matrices are given by the basic relations:

$\left[S_a,S_b\right]=i\varepsilon_{abc}S_c$

$\left[K_a,K_b\right]=-i\varepsilon_{abc}S_c$

$\left[S_a,K_b\right]=i\varepsilon_{abc}K_c$

Final remark: $SO(s,t)$ are sometimes called isometry groups since they define isometries over quadratic forms, that is, they define transformations leaving invariant the “spacetime length”.

# LOG#086. Group theory(VI).

We are going to be more explicit and to work out some simple examples/exercises about elementary finite and infinite groups in this post.

Example 1. Let us define the finite group of three elements as $(G,\circ)$ where $G=\left\{ I, M, M^2\right\}$, and such as the element $I_n$ and $M\in \mathcal{M}(n\times n)$ (matrices/arrays with n rows and columns, square matrices) with $M^3=I$ and where the matrix multiplication rule acts as group operation $\circ$. Then, this is an abelian group and its Cayley table can be written as follows:

 $\circ$ $I$ $M$ $M^2$ $I$ $I$ $M$ $M^2$ $M$ $M$ $M^2$ $I$ $M^2$ $M^2$ $I$ $M$

Example 2. Let $G$ be the following set of $2\times 2$ matrices:

$a=\begin{pmatrix}0 & -1\\ 1 & 0\end{pmatrix}$

$b=\begin{pmatrix}-1 & 0\\ 0 & -1\end{pmatrix}$

$c=\begin{pmatrix}0 & 1\\ -1 & 0\end{pmatrix}$

$d=\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix}$

If we choose the matrix multiplication as group multiplication, then the group $(G,\circ)$ is a group. The same set, with the operation “addition of matrices” $(G,+)$ is NOT a group. This is left as an exercise for you.

Example 3. Let $G$ denote the set of $2\times 2$ matrices with determinant equal to one (the unit) and complex “entries”. A generic element of this set is:

$X=\begin{pmatrix} a & b\\ c & d\end{pmatrix}$

It satisfies $\det X=+1$, where the determinant is defined to be (of course, a little of linear algebra is assumed to be known) $\det X= ad-bc=1$.

This group, with the ordinary matrix multiplication as group operation, is in fact a well defined group, since:

i) $\det (AB)=\det (A)\det B$

ii) $I$ is the identity matrix $\mathbb{I}=\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix}$

iii) $\forall A\in G\exists A^{-1}\in G$ since $\det A=1\Rightarrow \det A^{-1}=1$.

iv) The associative law holds for matrix multiplication and thus $(G,\circ)$ is a group (of matrices).

Example 4. The set $(\mathbb{Z},\circ)$, with $\circ$ defined by $n\circ m=n+m+1$ is a group. It can be easily proved checking that the neutral element is $e=-1$ and the inverse is $n^{-1}=-n-2$.

Example 5. The sets $(\mathbb{Z},+)$, $(\mathbb{Q},+)$, $(\mathbb{R},+)$, $(\mathbb{C},+)$ are abelian groups.

Example 6.  The set of every $2\times 2$ matrix with real or complex numbers with the operation + defined by

$\begin{pmatrix}a & b\\ c & d\end{pmatrix}+\begin{pmatrix}a' & b'\\ c' & d'\end{pmatrix}=\begin{pmatrix}a+a' & b+b'\\ c+c' & d+d'\end{pmatrix}$

is a group.

Example 7.  The “special” rotation group in two dimensions, commonly referred as $SO(2)$, is given by the matrices of the following class:

$R(\theta)=\begin{pmatrix}\cos \theta & \sin \theta\\ -\sin\theta & \cos \theta\end{pmatrix}$

This group is in fact abelian (something that it is NOT generally true with 3D or higher dimensional “rotation groups”). It can be easily proved that the composition of 2 different matrices of the above type is another matrix of the same type, and that the inverse or the unit element exist for every continuous and differentiable function f of two arguments. The neutral element is in fact $R(0)=I$ and $R(\theta)^{-1}=R(-\theta)$.

Example 8.   The 3D  “special” rotation group $SO(3)$ is defined by the set of the following 3 matrices (related to the celebrated Euler angles):

$R(\theta_1,\theta_2,\theta_3)=R(\theta_1)R(\theta_2)R(\theta_3)$

$R(\theta_1,\theta_2,\theta_3)=\begin{pmatrix}1 & 0 & 0\\ 0 & \cos\theta_1 & -\sin\theta_1\\ 0 & \sin\theta_1 & \cos\theta_1\end{pmatrix}\begin{pmatrix}\cos\theta_2 & 0 & -\cos\theta_2\\ 0 & 1 & 0\\ \sin\theta_2 & 0 & \cos\theta_2\end{pmatrix}\begin{pmatrix}\cos\theta_3 & -\sin\theta_3 & 0\\ \sin\theta_3 & \cos\theta_3 & 0 \\ 0 & 0 & 1\end{pmatrix}$

or equivalently R has the form

$\begin{pmatrix}\cos\theta_2\cos\theta_3 & -\cos\theta_2\sin\theta_3 & -\sin\theta_2\\ -\sin\theta_1\sin\theta_2\cos\theta_3+\cos\theta_1\sin\theta_3 & \sin\theta_1\sin\theta_2\sin\theta_3+\cos\theta_1\cos\theta_3 & -\sin\theta_1\cos\theta_2\\ \cos\theta_1\sin\theta_2\cos\theta_3+\sin\theta_1\sin\theta_3 &-\cos\theta_1\sin\theta_2\sin\theta_3+\sin\theta_1\cos\theta_3 & \cos\theta_1\cos\theta_2\end{pmatrix}$

This matrix is also important in neutrino oscillations and quark mixing. However, there the notation is a little bit different (beyond the fact that it has also extra “complex phases”). To simplify the notation and writing above, we can write $\sin\theta_i=s_i$ and $\cos\theta_j=c_j$ whenever $i,j=1,2,3$ (even the notation is useful with “extra dimensions” or higher dimensional rotation groups), so the above 3D matrix is rewritten as follows:

$R(1,2,3)=\begin{pmatrix}c_2c_3 & -c_2s_3 & -s_2\\ -s_1s_2c_3+c_1s_3 & s_1s_2s_3+c_1c_3 & -s_1c_2\\ c_1s_2c_3+s_1s_3 & -c_1s_2s_3+s_1c_3 & c_1c_2\end{pmatrix}$

In this case, the inverse element and the composition function are not “so easiy” as in the 2D case but they can be computed if you are patient and careful enough in a straightforward way. In particular, the composition rule for 3D rotations is NOT symmetrical in its arguments, and it shows that the group is non-abelian (although it has abelian subgroups, of course).

Example 9. The Lorentz group in 1D space and 2D spacetime is defined by the set of matrices ( with matrix multiplication and units $c=1$, i.e., natural):

$L(v)=\left\{\dfrac{1}{\sqrt{1-v^2}}\begin{pmatrix} 1 & -v\\ -v & 1\end{pmatrix}/ v\in (-1,1)\right\}$

This group, as it has OPEN intervals, and they are not closed, is “non-compact”. Non-compactness is an ackward property sometimes in physical/matheamtical applications, but it is important to know that feature. Compact groups are groups with parameters belonging to closed and bound/finite intervals (i.e., the interval limits can not be “infinite”). The Lorentz group in 2D spacetime IS abelian, and $L(0)=e=I$. However, it has a modified “cool” composition rule (the relativistic “addition” of velocities) given by the function:

$v_3=\dfrac{v_1+v_2}{1+v_1v_2}$

Moreover, the elements can be parametrized by a new parameter $\theta$, sometimes called “boost”, and defined as:

$\exp(\theta)=\dfrac{1}{\sqrt{1-v^2}}$

In terms of the boost parameter, Lorentz 2D spacetime transformations can be rewritten as follows:

$L(\theta)=\dfrac{1}{\sqrt{1-v^2}}\begin{pmatrix}1 & -v\\ -v & 1\end{pmatrix}=\begin{pmatrix}\cosh\theta & -\sinh\theta\\ -\sinh\theta & \cosh\theta\end{pmatrix}$

The composition rule with this parametrization is in fact very simple and it is related to hyperbolic trigonometry.

Example 10. Groups with less than 6 elements. We can eneumerate the finite groups with the help of the Lagrange’s theorem and/or building explicitly every possible multiplication table for a given number of elements. The results are given by the following list:

i) Using the Lagrange theorem, as 1,2,3 and 5 are prime, we do know that there is only one group of each class, the respective cyclic group with 1, 2, 3 and 5 elements.

ii) There exists at least one group of 4 elements: the cyclic group with 4 elements. In fact, there are two different groups with 4 elements:

ii.1) The so-called Klein’s group.

ii.2) The proper cyclic group with 4 elements.

Remark: We can enumerate easily every element in a cyclic group with n-elements. Using complex analysis we get that the n-elements in a cyclic group can be written with the following formula

$g_k=\exp \left(\dfrac{2\pi i k}{n}\right)$ $\forall k=1,2,\ldots,n$

Example 11. Congruences (I). Given $m\in \mathbb{Z}$ and numbers $x,y$ in that set, we ay that $x$ is congruent with $y$ modulus $m$, and we denote it by $x\equiv y(m)$ or even by $x=y(m)$ (sometimes it is writtem $\mbox{mod}(m)$ as well), if and only if:

$\exists k\in \mathbb{Z}/x-y=km$

This is an equivalence class/relation and we can define classes $\left[x\right]=\left\{y\in \mathbb{Z}/x\equiv y(m)\right\}$ and the set of classes

$\mathbb{Z}_m=\left\{\left[x\right],\left[y\right],\ldots\right\}$

Example 12. Congruences (II). We define an addition and a product acting on the congruence equivalente classes:

$(+):\left[a\right],\left[b\right]\in \mathbb{Z}_m\Rightarrow\left[a\right]+\left[b\right]=\left[a+b\right]$

$(\cdot):\left[a\right],\left[b\right]\in \mathbb{Z}_m\Rightarrow\left[a\right]\cdot \left[b\right]=\left[a \cdot b\right]$

With these operations it is possible to define in some cases groups, in the following way:

i) The group $(\mathbb{Z}_m,+)$. It is an abelian group $\forall m\in \mathbb{Z}$.

ii) The group $(\mathbb{Z}_m,\cdot)$ is a group if and only if (iff) $m$ is a prime number.

Example 13. Congruences (III). Write down the Cayley tables for $\mathbb{Z}_m$ if $m=1,2,3$.

Example 14. Permutation groups, sometimes denoted by $S_n$. This group is the (sub)group of the symmetrical group $S_n$ formed by the permutations of elements of a set of n-elements. Note that the group of all permutations of a set is in fact the definition of  the symmetric group. Then, the term permutation group is usually restricted to mean a subgroup of the symmetric group. The permutation group is the group formed by those elements that are permutations of a given set.

Example 15. The translation group in space. $T_{\vec a}T_{\vec b}=T_{\vec a+\vec b}$

In every vector space, it forms an (abelian) group.

Example 16. Translation group in spacetime. The same as before, but in spacetime. Translations and Lorentz groups in space-time together form the so-called Poincare group.

Example 17. Fractional linear transformations over the real numbers, the complex numbers or some other “beautiful and nice” class of numbers:

$A(x)=\dfrac{Ax+B}{Cx+D}$

where A,B, C and D are generally real, complex numbers or “similar numbers” with $AD-BC\neq 0$.

Example 18. Important matrix groups. Let A be the set of square matrices $n\times n$ over certain “field” $\mathbb{K}=\mathbb{R},\mathbb{C}$  (field said in the mathematical sense, not in the physical sense of “field”, be aware). The following sets with the usual matrix multiplication form a group with a continuous number of parameters (they are continuous infinite groups):

i) $G=GL(n,\mathbb{K})$. The general linear group. It is a continuous group with $n^2$ parameters. Generally, it is understood that the matrix is non-singlular in order to have a well-defined inverse element. Then, $\det M\neq 0\in G$.

ii) $G=SL(n,\mathbb{K})$. The special linear group. It has $n^2-1$ parameters over the field. They are the subgroup of $GL(n,\mathbb{K})$ with determinant equal to one.

iii) $G=U(n,\mathbb{C}$. The unitary group. It is formed by complex matrices that verify the property $MM^+=M^+M=I$, where $+$ denotes “adjoint” and transposition (the so-called hermitian conjugate by physicists). It has $n^2$ complex parameters, or $2n^2$ real parameters if you count in terms of real numbers.

iv) $G=SU(n)$. The subgroup of the unitary group formed with unitary matrices whose value is equal to the unit. $\det M=1$ in this group. Its number of parameters is given by $n^2-1$. The number of real parameters doubles it to be $2(n^2-1)$.

v) $G=O(n)$. The group of orthogonal matrices over the real numbers in euclidean space. It has $n(n-1)/2$ real parameters(generally “angles”). Any orthogonal matrix satisfies the property that $AA^T=A^TA=I$. The $SO(n)$ group is the subgroup of $O(n)$ formed by orthogonal matrices of unit determinant. The special orthogonal group has the same number of real parameters than the orthogonal group.

vi) $G=Sp(2n)$. The symplectic group. Sp(2n, K) is given by the set of 2n×2n matrices A (with entries in K) that satisfy

$\Omega A+A^T\Omega$ with

$\Omega=\begin{pmatrix} 0 & I_n\\ -I_n & 0\end{pmatrix}$

This version of the symplectic group is sometimes non-compact (and it is important in classical mechanics). There is another “symplectic” group, the group $Sp(n)$. It is the subgroup of quaternionic matrices GL(n,H), invertible quaternionic matrices (I am not going to explain the quaternions here this time). It is compact and formed by any matrix nxn over the quaternions leaving the hermitian form $x\cdot y=\overline{X}Y$. Then, the compact symplectic group can be related to the unitary group with some careful analysis on the number of parameters of the hermitian form.

May the group theory be with you!