Math 2.0

Group theory and generalizations Definition

Cayley table

Builds on Binary operation

Definition

Let S={a1,…,an}S = \{a_1, \ldots, a_n\} be a finite set with nn elements, listed in a fixed order, and let ∗\ast be a binary operation on SS. The Cayley table of (S,∗)(S, \ast) is the n×nn \times n table whose rows and columns are labelled a1,…,ana_1, \ldots, a_n. The entry in row ii and column jj is ai∗aja_i \ast a_j.

Remarks

The operation is commutative exactly when the table is symmetric about its main diagonal. In the Cayley table of a finite group, each element occurs exactly one time in each row and one time in each column. Thus the Cayley table of a finite group is a Latin square.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Cayley_table, Structure and layout
  2. Web https://en.wikipedia.org/wiki/Cayley_table, Properties and uses
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DefinesCayley table
UsesBinary operation

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