There is a direct link between groups represented by permutations and groups represented by permutation matrices. For example, if we have the permutation (12)(45) and we assume it belongs to S5, it can also be represented by the 5x5 matrix
0 1 0 0 0
1 0 0 0 0
0 0 1 0 0
0 0 0 0 1
0 0 0 1 0
In Sn, the conjugacy classes are all permutations with the same cycle structure, so the definition is necessary and sufficient. If we deal with subgroups of Sn, two conjugates must have the same cycle structure, but having the same cycle structure is not sufficient to state that the elements are conjugate. For example, the permutation (1234) generates a four element abelian group.
(1234)
(1234)² = (13)(24)
(1234)³ = (1432)
(1234)⁴ = (1)
(1234) and (1432) have the same cycle structure, but this group is abelian and every element is in a conjugacy class by itself. In S4, (24)(1234)(24) = (1432), but (24) isn't available here.
A group of matrices also has invariants under conjugacy, the determinant, which was discussed in the last post and the trace, which is the sum of the elements along the main diagonal. Let's look at a four element group of 2x2 matrices that represent rotations of 0°, 90°, 180° and 270° in the xy-plane. This is an abelian group isomorphic to the earlier permutation group, so every conjugacy class is a singleton.
0° matrix
1 0
0 1
determinant = 1, trace = 2
90° matrix
0 -1
1 0
determinant = 1, trace = 0
180° matrix
-1 0
0 -1
determinant = 1, trace = -2
270° matrix
0 1
-1 0
determinant = 1, trace = 0
Note that the 90° matrix and the 270° matrix have the same determinant and trace, which means there is a 2x2 non-singular matrix M such that M(90° matrix)M⁻¹ = 270° matrix. One such matrix is
-1 0
0 1
which is not one of the elements of our defined group.
Commentary
As I have stated earlier, group theory is a generalization of the concept of symmetry. It is taught as an abstract field and the applications seem remote, with the possible exception of Rubik's Cube solutions. In fact, having symmetry in a physical problem makes it easier to solve. According to my professor Stu Smith, all solved differential equations rely on symmetry except for one. The differential equation that solves the solitary wave, also known as a soliton, does not have symmetry.
Most waves have peaks and valleys, and when the hit other waves, they can add to each other or cancel each other out, but a soliton only is almost all peak with a tiny valley. The bigger a soliton is, the faster it moves. In the ocean, we call the biggest solitons tsunamis.
A basic tenet of physics is big + fast = fuck you up. This is why a tsunami can wreak havoc when it hits land thousands of miles away from the source, because it will nearly the same speed and size it had when it was formed.
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