Cayley Table

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tags: - colorclass/functional analysis ---see also: - Representation Theory

A Cayley table, also known as a group table, is a matrix-like representation of a group that shows the result of the group operation applied to every pair of elements from the group. Named after the mathematician Arthur Cayley, this table is a useful tool for studying small groups and understanding their structure, particularly for finite groups.

Definition and Structure

A Cayley table for a group $G$ with $n$ elements is an $n\times n$ table where each row and column corresponds to an element of $G$. The entry in the row labeled $a$ and column labeled $b$ is the group operation of $a$ followed by $b$, denoted as $ab$. If the group operation is denoted multiplicatively, this is analogous to multiplication; if it’s denoted additively (as in an additive group), this corresponds to addition.

Properties

1. Closure: By definition, since $G$ is a group, the operation on any pair of elements from $G$ results in another element from $G$. 2. Associativity: While not directly evident from the Cayley table itself, the group operation is associative. Hence, the order in which operations are performed in a sequence does not affect the outcome. 3. Identity Element: There exists an element $e$ in $G$ such that for any element $a$ in $G$, the operations $ea=ae=a$. In the Cayley table, this means one row and one column will effectively repeat the labels of the rows and columns. 4. Inverses: For every element $a$ in $G$, there exists an inverse element $a^{-1}$ such that $a{a}^{-1}=a^{-1}a=e$. In the table, this means that every row and column will contain the identity element $e$.

Example: Cayley Table for ${\mathbb{Z}}_3$

Consider the additive group ${\mathbb{Z}}_3=\{0,1,2\}$ where the operation is addition modulo 3. The Cayley table is structured as follows:

$+$	0	1	2
0	0	1	2
1	1	2	0
2	2	0	1

This table illustrates that ${\mathbb{Z}}_3$ is a cyclic group generated by $1$, as repeated addition of $1$ (modulo 3) results in every element of the group.

Applications

Cayley tables are primarily used in group theory to:

- Verify if a small set and operation form a group. - Understand the structure of the group, such as whether it is abelian (commutative), by checking if the table is symmetric along the diagonal. - Help in visualizing and identifying subgroups, normal subgroups, and conjugate elements within a group.

Limitations

While Cayley tables are incredibly useful for small groups, their practicality diminishes as group size increases due to the exponential growth of the table’s size. For larger groups, abstract properties and theorems become more practical tools for analysis than direct computation or table visualization.

In conclusion, the Cayley table serves as a fundamental and illustrative tool in group theory, enabling detailed examination and understanding of group operations and structures, especially in finite groups.