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

In summary, the conversation discusses the relationship between a Cayley table of a finite set and whether or not the set is a group. It is mentioned that the converse is true and that inverses exist and are unique. While there is a row and column representing the identity, it is uncertain if they are the same element and if the structure is associative. However, it is noted that the array does not necessarily represent an associative structure, as shown in an example on the Wikipedia article on Latin Squares.
  • #1
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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  • #2
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.
 

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

What is a Cayley Table?

A Cayley Table is a visual representation of the operation table for a particular mathematical structure, such as a group or a ring. It displays all possible combinations of elements and their resulting operations.

What is a Group?

A Group is a mathematical structure that consists of a set of elements and an operation that combines any two elements in the set to produce a third element. The operation must also satisfy certain properties, such as closure, associativity, identity, and invertibility.

What are the properties of a Group?

The properties of a Group are closure, associativity, identity, and invertibility. Closure means that the operation on any two elements in the group will result in a third element that is also in the group. Associativity means that the order in which the operations are performed does not matter. Identity means that there is an element in the group that, when combined with any other element, will result in that element. Invertibility means that for every element in the group, there exists an inverse element that, when combined with the original element, will result in the identity element.

How is a Cayley Table used to represent a Group?

A Cayley Table is used to represent a Group by organizing the elements of the group into rows and columns. The operation is then performed on the elements in each row and column, and the resulting elements are placed in the corresponding cell in the table. This allows for a visual representation of the operation and the relationships between elements in the group.

What is the significance of the Cayley Table in Group theory?

The Cayley Table is significant in Group theory because it allows for a visual representation of the operation and the properties of a Group. It also helps to identify patterns and relationships between elements in the group, which can lead to a better understanding of the group's structure and properties.

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