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3 - Third axiom of groups' theory :
Any element of the group must own its inverse, written g⁻¹, defined by :
(15) g × g⁻¹ = g⁻¹ × 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 × g₂) = det (g₁) × det (g₂)
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 × g⁻¹ = g⁻¹ × g = 1
Then :
(19)
det ( g × g⁻¹ ) = det (g) × 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 × g₂ ) × g₃ = g₁ × ( g₂ × 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 G is 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 × 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.