In an earlier blog we examined the fascinating mathematical structures called groups. We recall here the properties that make a group, from the definition given:

A GROUP is a set G with one binary operation defined on it , and G satisfies the axioms:

a) Associative law: a· (b · c) = (a· b) · c

b) Identity element (e): there exists an element e in G such that: e · a = a for all a in G and a · e = a

c) Inverse element: For any a in G there exists an element a^-1 in G such that:

a · a^-1 = a^-1 · a = e

There was also the additional property that defined an Abelian Group:

d) Only if there exists elements a, b in G such that (a · b) = (b · a), then G is said to be an

We now want to see how these apply to groups that are posed in matrix form.

A matrix is an assembly of numerical quantities in the form:

(a11 a12)

(a21 a22)

Matrix multiplication, the main operation we will need to know, is easily obtained.

I.e. say the above matrix is denoted A, and another B =

(b11 b12)

(b21 b22)

then A X B =

(a11 a12) (b11 b12)

(a21 a22) (b21 b22)

= [{(a11b11) + (a12b21)} --{(a11 b12)+ (a12 b22)} ]

[(a21b11) + (a22b21) } --{((a21 b12) + (a22 b22)}]

For example, let A=

(1 2)

(1 2)

and B =

(1 3)

(2 2)

then A x B=

(5 7)

(5 7)

The reader is invited to verify this for himself.

We are now in a position to look at some simple 2 x 2 matrix groups, and also assess whether they're Abelian or not.

One example is

The elements are shown in their standard matrix form in Fig. 1, and the reader should easily be able to verify that the elements form a group. I tcan be seen, for example, that: II= J*J= J^2 = K*K = K^2 = -i, and IIJ = -JII = K.

Another interesting group is PSL(2, z) which has generators:

s =

(0 1)

(-1 0)

and

t =

(0 -1)

(1 - 1)

Yet another group with 2 x 2 matrix elements is sl(2) which has elements: h, e and f such that:

h =

(1 0)

(0 -1)

e (identity element) =

(0 1)

(0 0)

f =

(0 0)

(1 0)

The elements of the group can easily be shown to obey the relations:

[h, e] = h*e - e*h = 2e, [h.f] = h*f - f*h = -2f, and [e,f] = e*f - f*e = h

(understanding that matrix subtraction simply follows the rule, e.g. :

[A] - [B]

=

{(a11 - b11) (a12 - b12)}

{(a21 - b21) (a22 - b22)}

using the designated elements assigned earlier for the generic matrices A, B)

For the already identified matrices A and B, [A] - [B] =

(0 -1)

(-1 0)

Then there is the famous Klein Viergruppe with members: e (identity), a, b and c. The 2 x 2 matrix members are shown in Fig. 2 along with 4 different operations.

The ambitious reader can gain further insights via the following exercises!

Practice Problems:

1. For the group PSL(2,z) show that the identity element (e) = s^4 = t^3.

2. For the group sl(2) show that:

(a) [h.f] = h*f - f*h = -2f

(b) The "Casimir element", C, of sl(2) is defined according to:

h^2/ 2 + h + 2f*e

find the element

3. Show that the Klein Viergruppe, V4, is

A GROUP is a set G with one binary operation defined on it , and G satisfies the axioms:

a) Associative law: a· (b · c) = (a· b) · c

b) Identity element (e): there exists an element e in G such that: e · a = a for all a in G and a · e = a

c) Inverse element: For any a in G there exists an element a^-1 in G such that:

a · a^-1 = a^-1 · a = e

There was also the additional property that defined an Abelian Group:

d) Only if there exists elements a, b in G such that (a · b) = (b · a), then G is said to be an

*Abelian**Group.*(Commutative property).We now want to see how these apply to groups that are posed in matrix form.

A matrix is an assembly of numerical quantities in the form:

(a11 a12)

(a21 a22)

Matrix multiplication, the main operation we will need to know, is easily obtained.

I.e. say the above matrix is denoted A, and another B =

(b11 b12)

(b21 b22)

then A X B =

(a11 a12) (b11 b12)

(a21 a22) (b21 b22)

= [{(a11b11) + (a12b21)} --{(a11 b12)+ (a12 b22)} ]

[(a21b11) + (a22b21) } --{((a21 b12) + (a22 b22)}]

For example, let A=

(1 2)

(1 2)

and B =

(1 3)

(2 2)

then A x B=

(5 7)

(5 7)

The reader is invited to verify this for himself.

We are now in a position to look at some simple 2 x 2 matrix groups, and also assess whether they're Abelian or not.

One example is

*the special unitary group*, SU2. The elements of SU2 are the unitary 2 x 2 matrices with Det = 1 (determinant). [Note: the determinant is taken as follows, using the elements of matrix A, for which Det [A] = (a11 x a22) - (a12 x a21), thus Det [A] = 2 - 2 = 0.]The elements are shown in their standard matrix form in Fig. 1, and the reader should easily be able to verify that the elements form a group. I tcan be seen, for example, that: II= J*J= J^2 = K*K = K^2 = -i, and IIJ = -JII = K.

Another interesting group is PSL(2, z) which has generators:

s =

(0 1)

(-1 0)

and

t =

(0 -1)

(1 - 1)

Yet another group with 2 x 2 matrix elements is sl(2) which has elements: h, e and f such that:

h =

(1 0)

(0 -1)

e (identity element) =

(0 1)

(0 0)

f =

(0 0)

(1 0)

The elements of the group can easily be shown to obey the relations:

[h, e] = h*e - e*h = 2e, [h.f] = h*f - f*h = -2f, and [e,f] = e*f - f*e = h

(understanding that matrix subtraction simply follows the rule, e.g. :

[A] - [B]

=

{(a11 - b11) (a12 - b12)}

{(a21 - b21) (a22 - b22)}

using the designated elements assigned earlier for the generic matrices A, B)

For the already identified matrices A and B, [A] - [B] =

(0 -1)

(-1 0)

Then there is the famous Klein Viergruppe with members: e (identity), a, b and c. The 2 x 2 matrix members are shown in Fig. 2 along with 4 different operations.

The ambitious reader can gain further insights via the following exercises!

Practice Problems:

1. For the group PSL(2,z) show that the identity element (e) = s^4 = t^3.

2. For the group sl(2) show that:

(a) [h.f] = h*f - f*h = -2f

(b) The "Casimir element", C, of sl(2) is defined according to:

h^2/ 2 + h + 2f*e

find the element

3. Show that the Klein Viergruppe, V4, is

*Abelian.*

*Solutions to be given in a future blog!*
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