SIMPLICITY OF THE SO3 GROUP

ABSTRACT

The article is devoted to the theory of groups and specifically to the SO3 group. Discussion of the simplicity of the SO3 group, analysis of examples of tasks, as well as the scope of their application is the basis for our discussion. The result of this discussion is the systematization of the accumulated material on this issue, as well as the compilation of an idea about the main directions of modern group theory, about its methods, about its greatest achievements.

Keywords: SO3 group, group theory, mathematics, rotation group.

Group theory has a long and meaningful history. Originated in connection with Galois theory and for the needs of this theory, it developed first as a theory of finite permutation groups (Cauchy, Jordan, Silov).

In the future, the work in the general theory of groups became more and more turbulent and versatile, and by now this part of mathematics has turned into a broad and rich in content science, occupying one of the first places in modern algebra. It is clear that this development of the general theory of groups could not ignore the successes already achieved in the theory of finite groups. On the contrary, much of this development arose from the corresponding parts of the theory of finite groups, and the guiding one was the desire to replace the finiteness of the group with those natural constraints under which this theorem or this theory still remains valid and beyond which they lose their force.

The theory of groups is far from complete. The multiplicity of specific problems facing it, as well as the presence of areas in which work has begun only recently, allow us to believe that the general theory of groups has not yet passed through the peak of its development.

It is quite timely, however, to systematize the rich material that has already accumulated and thereby give a wide circle of mathematicians an idea of the main directions of modern group theory, its methods, and its greatest achievements.

The SO3 group is a special orthogonal group of dimensions 3. It belongs to the rotation group. By definition, all orthogonal transformations of three-dimensional space, with orientation preserved (right and left three vectors).

In the orthonormal basis, transformation matrices from SO3 are orthogonal matrices (А^T=A^-1) with determinant 1.

In the course of linear algebra, it was proved that the matrix of an element from SO3 can be reduced in an orthonormal basis to the form:

That is, any SO3 element is a rotation around some axis.

Lemma. Let  - turning the corner Ө around the axis I. Then the conjugate element ghg^-1 - this is a rotation by an angle Ө around the axis g(I).

The conjugate operator has the same complex eigenvalues. This proves the equality of angles. It remains to explain that the axis of rotation will be g(I).

Let v - the eigenvector of the operator hс has an eigenvalue of 1. Then (ghg^-1) (gv)=ghv=gv, so gv - an eigenvector with an eigenvalue of 1 for the operator ghg^-1.

Lemma. Composition of two turns at an angle πc the angle between the axes I and I' equal to α is equal to the rotation relative to the axis m perpendicular to I and I' by the angle 2α.

The m axis turns over at each of these rotations, which means that as a result, an identical transformation occurs with the m axis. The I axis remains stationary at the first turn, and at the second it rotates at an angle of 2α.

The theorem. The SO3 group is simple.

Let H≠{id} - a normal subgroup in SO3. There will be a turn h∈H on the corner α∈(0,2 π) around the axis I. Let g be a rotation by an angle π around the axis m forming an angle with the axis I ꞵ∈[0, π/2]. Then s=g(hg^-1h^-1)=(ghg^-1)h^-1∈H. In this case, hg-1h-1 is a rotation by an angle π around the h(m) axis, which forms a mughol γ with the axis. This means that s is a rotation by an angle of 2y around an axis perpendicular to mi h(m).

The angle γ is equal to 0 at ꞵ=0 and is equal to α at ꞵ= π/2.For reasons of continuity, the angle γ can take all values from 0 to α, that is, there will be rotations at all angles from 0 to α. Considering the degrees of these turns, we get turns at all angles.According to the lemma, if there is a turn at a certain angle, then all turns at a given angle lie there. So H=SO(3).

Consider the action of a group G on the set of all subgroups in the group G by conjugates. Indeed, it is easy to see that if H⊂G is a subgroup, then gHg -1 is also a subgroup.

Definition.

The stabilizer of the subgroup H in this action is called the normalizer Hb G and is denoted NG(H).

Lemma.

1. NG(H) is a subgroup in G containing H.
2. H is normal in NG(H).
3. If H is normal in K, where K is a subgroup of G, then K∈N_G(H).

Proof.

1) By definition, NG(H) is a stabilizer, which means a subgroup in G.

2) For g∈NG(H) we have gHg−1 = H, this proves the normality of H in NG(H).

3) If H / K, then for any k∈K, kHk−1 = H is fulfilled, that is, k∈St(H) = NG(H).

Thus, the non-abelicity of the rotation group is associated with non-zero structural constants. It is clear that if the commutator of infinitesimal rotations (generators) is different from zero, then the matrices of arbitrary rotation will not commute among themselves [3]. Naturally, Abelian subgroups can also be distinguished in the group SO(3). For example, rotation around one axis at different angles is an Abelian group SO(2)∈SO(3). The description of the composition of rotations is greatly simplified when using the quaternion technique.

References:

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2. Barut, A., Ronchka, R. Representation theory of groups and its applications. Volumes 1-2. Moscow: Mir, 1980.
3. Weil, G. Classical groups, their invariants and representations.-M.: Gosizdat, 1947. - 48c.
4. Vinberg, E.B. Linear representation of groups. - M.: Nauka, 1985. -144c.
5. Dick, T. Transformation groups and representation theory. - M.: Mir, 1982. - 227s.
6. Kargapolov, A.I., Merzlyakov Yu.I. Fundamentals of group theory. - M.: Nauka, 1982.-288s.
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