Multiplication table of a finite group

This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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Definition

Definition with symbols

Let $G$ be a finite group of order $n$ . Then, the multiplication table of $G$ is a $n \times n$ matrix described as follows:

Label the elements of $G$ as $g_{1}, g_{2}, \dots, g_{n}$ such that $g_{1}$ is the identity element.
Now, in the $(i, j)^{t h}$ entry of the matrix, put the element $g_{i} g_{j}$

Note that under this definition, the multiplication table is an $n \times n$ matrix with entries drawn from $G$ . We can, however, change this to a $n \times n$ matrix with entries drawn from $1, 2, \dots, n$ by simply replacing $g \in G$ by the $k$ for which $g = g_{k}$ .

The latter interpretation makes the multiplication table into a purely combinatorial object. We shall use this latter interpretation.

Clearly, the multiplication table determines the group uniquely.

We can also construct the multiplication table of any finite magma in the same way.

Properties

Latin square

The multiplication table of a finite group is a Latin square, when we view it as having entries $1, 2, \dots, n$ . In other words, every row contains each element exactly once and every column contains each element exactly once.

In fact, the multiplication table of a finite magma is a latin square if and only if that magma is a quasigroup. Groups are indeed quasigroups.

Since we chose $g_{1}$ as the identity element, the first row and first column are both $1, 2, \dots, n$ .

Symmetric Latin square

The multiplication table of a finite group (and more generally, of any finite magma) is a symmetric matrix if and only if the binary operation is commutative. In the case of groups, this boils down to saying that the group is Abelian.

Automorphisms of the group

Let $P$ be the permutation matrix corresponding to an automorphism of $G$ . If $M$ denotes the multiplication table matrix for $G$ , we have:

$P M P^{t} = M$

This is equivalent to saying that $M$ commutes with $P$ .

In other words, the centralizer of $M$ in $G L_{n} (Z)$ contains all permutation matrices arising from automorphisms of $G$ .