Is a Cayley Table a Reliable Indicator of a Finite Set Being a Group?

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SUMMARY

The discussion centers on the relationship between Cayley tables and the properties of finite sets being groups. It is established that if a Cayley table of a finite set G is a Latin square, it does not necessarily imply that G is a group, as associativity is not guaranteed. The conversation references the existence of quasigroups with identity, which serve as counterexamples to the assumption that a Latin square structure ensures group properties. The need for a proof or counterexample regarding associativity is highlighted.

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  • Understanding of Cayley tables and their properties
  • Familiarity with Latin squares and their definitions
  • Knowledge of group theory, particularly the definitions of groups and quasigroups
  • Basic concepts of mathematical proofs and counterexamples
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  • Study the definitions and examples of quasigroups and their identities
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Bleys
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If the Cayley table, of a finite set G, is a latin square (that is, any element g appears once and only once in a given row or column) does it follow G is a group?
I know the converse is true, and it seems reasonable that this is true. Since the array will be of size |G|x|G|, inverses exist and are unique. There will be one row (and one column) that will represent the multiplication by the identity (since the rows and columns are permutations of the set). But I don't know about associativity; in fact I'm not even sure the if the column and row representing the identity will actually be the same element. So is there a counter-example or if it's true, a proof?
 
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Never mind, the array need not represent an associative structure. There is an example on the wikipedia article on Latin Squares that shows one of a quasigroup with identity.
 

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