Cayley's theorem
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Cayley's theorem
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Let Sn be the symmetric group on n elements Any finite group is isomorphic to a subgroup of Sn for some n. Proof [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] Now suppose [Equation goes here - download the original to see it.] where e is the identity permution in Sn. [Equation goes here - download the original to see it.] This means that the kernel of the homomorphism is the identity. In symbols: [Equation goes here - download the original to see it.]. Hence, G is isomorphic to a subset of [Equation goes here - download the original to see it.]. Illustratio To illustrate this proof, consider the group of symmetries of the square. These are 1.[Equation goes here - download the original to see it.] 2. [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] 3. Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] 4 [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] 5. [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] 6. [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] 7. [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] 8. [Diagram goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] Note, in constructing this table there is no real alternative to just working out most of the entries by hand, the long way. For example, [Equation goes here - download the original to see it.] The mappings, tg are given by converting the rows of this table into functions [Diagram goes here - download the original to see it.] This exhibits each row as a mapping. The general idea of the mapping can be seen from the following. [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] Each mapping is a permutation of the 8 elements of G, and there are 8 mappings in all. However, in S8 there are 8! = 40320 such permutations, and hence 40320 such mappings. That is rather a lot of mappings, and shows that in some ways Cayley's theorem is interesting only from a theoretical point of view. Since the permutation group in which any other group is embedded is very much larger than that group, this does not tell us much new about its structure. However, it is possible to find a smaller Sn in which to embed a group. In the case of the symmetries of the square, this is S4, which is the permutation group of four elements, and also the symmetry group of the tetrahedron. There are 4! = 24 elements in S4 [Diagram goes here - download the original to see it.]
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Contents of
Cayley's theorem
1 Cayley's theorem
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