| 2 |
3 - Third axiom of
groups' theory :
Any element of the group
must own its inverse , written g-1 , defined by :
(15)
g
x g-1 = g-1
x g = 1
In our example :
(16)
i.e : b
= - a or :
(17)
g-1
( a ) = g ( - a )
Here the calculation
of the inverse matrix is trivial.
What is the condition
for a given square matrix to own its inverse ?
...To
any square matrix we can associate a scalar called determinant . For
definition see any book devoted to linear calculus . This determinant
is codified : det ( g )
In addition we have a
general theorem :
det (g1
x g2)
= det (g1 )
x det (g1 )
The determinant of a
diagonal matrix is :
(18)
As a consequence :
det ( 1 ) = 1
for 1 is a diagonal matrix.
From the definition of
the inverse of a matrix :
g
x g-1 = g-1
x g = 1
Then :
(19)
det ( g
x g-1 ) = det (g)
x det (g-1 ) = 1
...If
det (g) = 0 the condition (19) cannot be satisfied. Sets of matrixes
whose peculiar elements own null determinant don't satisfy the third axiom,
and cannot form a group.
By the way :
(20)
4 - Fourth axiom
of groups'theory :
The multiplication must
be associative, i. e : .
(21)
( g1
x g2
) x g3
=g1
x ( g2
x g3 )
Matrix multiplication
is basically associative.
Dimension of a group
:
...As
we will see, a group may act on a space whose points are described by column-vectors.
For an example space-time points (called "events") :
(22)
...This
is a four dimensions space. Different groups may act on it. But the dimension
of a group has nothing to do with the dimension of the space it acts on.
The dimension of
a group (of matrixes) is the number
of parameters which
define these square matrix.
We have given an example
of matrixes, defined by a single parameter
a
So that the dimension
of this group is one.
Notice that :
(22-bis)
Remark :
All groups of matrixes
are not commutative , although the group we studied owns this property
:
(23)
If such group acts on a
column vector, corresponding to a 2d space :
(23 bis)
it corresponds to rotation around a fixed point, in a plane :
(23 ter)
This operation is obviously
commutative.
You will tend to say
: "like all rotations groups".
...You're
wrong. Consider the rotations around axis passing by a given point O. Combine
two successive rotations, around different axis. This is not commutative.
Exercise : show that, using orthogonal axis system (OX, OY, OZ), combined
rotations around these axis is not a commutative operation.. Take any object.
- Make a rotation +90°
around OX, then a rotation +90° around OZ
Return to initial conditions
and :
- Make a rotation +90°
around OZ, then a rotation +90° around OX
Compare the results.
Group's action.
...A
group Gis composed by square matrixes g . They can be multiplied.
We will say that a group may act on itself .
The group may also act on a space, made of points, described by column vectors.
Example :
(24)
If we write :
(25)
the action of the group
on this space becomes :
(26)
g
x r
...In
this peculiar case the action on space identifies to the simple matrix multiplication.
But the concept of an action is much more general.
Index Dynamic Groups Theory