Definition:Inverse Row form of Cayley Table for Group

Definition

Let $\struct {G, \circ}$ be a finite group.

The Cayley table for $\struct {G, \circ}$ can be presented in a form such that the rows are headed by the inverse elements of the elements which head the corresponding columns.

This form is known as the inverse row form of the Cayley table for $\struct {G, \circ}$.

Examples

Symmetric Group on $3$ Letters

The Cayley table of the symmetric group on $3$ letters can be written in inverse row form as:

$\begin{array}{c|cccccc}

\circ & e & p & q & r & s & t \\ \hline e & e & p & q & r & s & t \\ p^{-1} & q & e & p & t & r & s \\ q^{-1} & p & q & e & s & t & r \\ r^{-1} & r & t & s & e & q & p \\ s^{-1} & s & r & t & p & e & q \\ t^{-1} & t & s & r & q & p & e \\ \end{array}$

Also see

• Results about Inverse Row form of Cayley Table for Group can be found here.

Sources

• 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 5$: The Multiplication Table