Identity Elements occupy Diagonal of Cayley Table in Inverse Row Form

Theorem

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

Let $\CC$ be a Cayley table for $\struct {G, \circ}$ presented in inverse row form.

Then all the entries in the main diagonal of $\CC$ are instances of the identity element.

By definition of inverse row form, the rows of $\CC$ are headed by the inverse elements of the elements which head the corresponding columns.

The entries in the main diagonal of $\CC$ have the same column number as row number.

Let $\sqbrk c_{k k}$ denote the entry of $\CC$ corresponding to the element where the $k$th row intersects the $k$th column.

Let $a$ be the element which heads column $k$.

Then, by definition, $a^{-1}$ is the element which heads row $k$.

Thus the element which occupies entry $\struct c_{k k}$ is $a^{-1} \circ a = e$.

Hence the result.

$\blacksquare$

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