Introductory Modern Algebra by Stahl Saul

Author:Stahl, Saul
Language: eng
Format: epub
ISBN: 9781118552032
Publisher: Wiley
Published: 2013-09-27T04:00:00+00:00

9.6 Cayley’s Theorem

The first groups to be examined by mathematicians were groups of permutations. It was not until a century had past that Cayley pointed out that every group is determined up to isomorphism by its multiplication table, and that therefore this table could be used to define the notion of an abstract group. At the same time Cayley noted that this innovation did not introduce any genuinely new structures into the study of groups, for, he said, every abstract group can be shown to be isomorphic to a group of permutations. Cayley did not formally prove this assertion; he contented himself with an example. His short note on the subject is included as Appendix E. Cayley’s assertion will be formally stated and proved below as Theorem 9.18, but we first paraphrase Cayley’s ideas in more modern terminology. Table 9.12 contains the multiplication table of the symmetric group S3 with a = (1 2), b = (3 2 1), c = (1 3), d = (1 2 3), and e = (2 3).

Table 9.12 The multiplication table of S3

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