# 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!

# LOG#084. Group theory (IV).

Today we are going to speak about two broad and main topics: cyclic groups and some general features of finite groups (a few additional properties and  theorems).

A cyclic group is, informally speaking, a group that can be generated by a single element, the group has an element $g$ that we can call “the generator” of the group and we can “make” every element of the group by direct multiplication or “powering” the element g (or by multiplication by g once and once and again, in an additive language).

Definition (17). Cyclic group. We say a group $(G,\circ)$ is cyclic if

$\exists x\in G/G=\left\{e,x,x^2,\ldots,x^{n-1}\right\}$

The above group is at most a group with n-elements, and x is said to be “the generator” of the cyclic group.

Propostition. If $(G,\circ)$ is cyclic, then G is abelian.

Check: Let $a\in G$ with $a=x^p$ and let $b\in G=x^q$. Then

$a\circ b=x^p\circ x^q=x^{p+q}=x^{q+p}=x^q\circ x^p=ba$ QED

By the way, the product symbol/binary operation will be omited with frequency when no confusion in the composition of elements is possible.

Cyclic groups some have beautifuld and interesting features:

1st. The fundamental theorem of cyclic groups asis that every subgroup of a cyclic group is cyclic.

2nd. The order of any subgroup of a finite cyclic group of n elements is a divisor of n.

3rd. For every positive divisor k of n the group G has exactly one subgroup of order k.

This last property characterizes finite cyclic groups: a group of order n is cyclic if and only if for every divisor $d$ of $n$, the group has at most one subgroup of order $d$.

Proposition. Let G be a finite group, given $x\in G$ exists $n\in\mathbb{N}/x^n=e$.

Check: Given $x\in G$ if there is not a $n\in \mathbb{n}$ such that verifies that property, then there are some numbers $m', n' \in \mathbb{N}$ such that $x^{n'}=x^{m'}$. If not, the group is NOT finite! Therefore, if $n=n'-m'$, then we have

$x^n=x^{n'}x^{-m'}=e$

That is a contradiction and the proposition truth follows in a straightforward manner.

Proposition. Let G be a finite group. Let us define

$x\in G,n=\left\{\mbox{min}(m)/x^m=e\right\}$

and

$A_x=\left\{e,x^{n-1},\ldots,x\right\}$

Then, every element of $A_x$ is different.

Check: It there are 2 elements $x^p=x^q$ with $p,q then $x^{p-q}=e$ and $p-q, so n is not the minimum.

These features allow the following definition:

Definition (18). Given a finite group, and the element $x\in G$, the order of x is defined to be

$\mbox{order}(x)=\mbox{min}\left\{m/x^m=e\right\}$

Theorem (Lagrange). Let G be a finite group, then $\forall x\in G$ we have

$\dfrac{\vert G\vert}{\mbox{order}(x)}=n\in \mathbb{N}$

i.e., the natural number $\mbox{order}(x)$ is a divisor of $\vert G\vert$, the number of elements of G.

Proposition. Let $H\subseteq G$, $(H,\circ)$ be a subgroup of $(G,\circ)$, then the number of elements of H, $\vert H\vert$ is a divisor of $\vert G\vert$.

Some consequences of the Lagrange theorem are the following propositions:

Proposition. If $(G,\circ)$ is a group, and $\vert G\vert =p$, where $p$ is a prime number, then G is a cyclic group.

Proposition. For all $n\in \mathbb{N}$ exists, at least, the cyclic group with $n$ elements.

We will write from now $(G,\circ)$ as G for shorthand notation.

Definition (19). Conjugate elements and classes of conjugacy.

Let $x,y\in G$ two elements in G. They are said to be conjugated $x\sim y$ if exists an element $h\in G$ such as

$x=hyh^{-1}$

This definition provides an equivalence relation in G. The equivalence classes $\sim$ of conjugate elements in a group G are called “conjugacy classes” of G.

Definition (20). Conjugate elements for a subgroup. If G is a group and H is a subgroup of G, we ay that two elements $x,y\in G$ are related through H, and we write $x\equiv y(H)$ if $x^{-1}y\in H$.

Proposition. Let G a group and H a subgroup of G. Then,

a) The relation of conjugation defined above is an equivalence relation.

b) The equivalence class of $x\in G$ in this relation are in correspondence with the set $xH$, the product of elements of G by those in H.

Proposition. Let G be a group and H a subgroup. Then given an element $x\in G/x\ni H$, the set

$A=xHx^{-1}=\left\{xhx^{-1}/h\in H\right\}$

is a subgroup of G.

Definition (21). Conjugate group. Given $H\subseteq G$ a subgroup of G, and an element $x\in G$ then the subgroup

$A=xHx^{-1}$

is called the conjugate subgroup of H. Alternatively, A is conjugated with H if and only if (iff)

$\exists x\in G/xH=Ax$

See you in the next group theory blog post.