Permutation groups
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Permutation groups
Equations are omitted for technical reasons - download the original pdf
A permutation is an arrangement of numbers or objects in a specific order. For example there are six permutations of the numbers, 1,2 and 3 as follow. [Equation goes here - download the original to see it.] Group theory deals with the structures created when a binary operation is defined on a set in such a way as to satisfy the group structure properties of (1) existence of identities, (2) existence of inverses; (3) closure of the group; (4) associativity of the operation. An example of a group is the symmetry group of the equilateral triangle, which shall be discussed below. There are six transformations that can be described as acting on an equilateral triangle so that the triangle is left unchanged at the end of it - these are basically reflections in certain axes and certain rotations about the centre. These transformations form a set, and the binary operation of composition of transformations makes that set into a group. Hence the symmetry group of the equilateral triangle. However, mathematicians are always looking for ways of turning one structure into another. It is useful to manipulate numbers rather than transformations. One approach is to label the vertices of, in this case, the equilateral triangle, and to examine how the various transformations of the symmetry group affect these triangles. This is only an example, but it leads us to the idea in general of a permutation group - that is a group defined on a set of numbers, made up of the collection of all permutations of those numbers. Let us examine in detail how this works for the symmetry group of the equilateral triangle.
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Contents of
Permutation groups
1 Permutation groups
2 Symmetry groups are also permutation groups
3 Symmetry group of the square
4 Rotation through phi
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