On 30 November 1812, the young engineer from Cherbourg read to the Institut de France a memoir with a long title: on functions that can take only two values, equal and of opposite signs, when the variables are interchanged. Behind the title was the first general theory of determinants.

Functions that change sign

Take three variables $a_1, a_2, a_3$ and the product of the differences

$$\Delta = (a_2 - a_1)(a_3 - a_1)(a_3 - a_2).$$

If we swap $a_1$ and $a_2$, each factor either stays the same or changes sign, and altogether $\Delta$ becomes $-\Delta$. Any swap of two variables does the same. A permutation of the variables, which is a sequence of swaps, multiplies $\Delta$ by $+1$ or by $-1$: it is even or odd.

Cauchy used this observation to organise the expressions that Leibniz, Cramer, Vandermonde and Laplace had found when solving linear systems. He laid out the $n^2$ coefficients in a square array with two indices, $a_{i,j}$, and defined the determinant as the sum of all products with one entry from each row and each column, each with the sign of the corresponding permutation:

$$\det\begin{pmatrix} a_{1,1} & a_{1,2} \\ a_{2,1} & a_{2,2}\end{pmatrix} = a_{1,1}a_{2,2} - a_{1,2}a_{2,1}.$$

The word “determinant” came from Gauss, who had used it for quadratic forms; it was Cauchy who fixed it in today’s sense.

The product

The main result of the memoir is the product theorem: if $C$ is the array obtained by multiplying $A$ by $B$ row by column, then

$$\det C = \det A \cdot \det B.$$

Jacques Binet presented the same theorem that same year, and the general version, for rectangular arrays, is now called the Cauchy–Binet formula. Here it is with numbers. With

$$A = \begin{pmatrix} 2 & 1 \\ 1 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 4 \\ 2 & 1 \end{pmatrix},$$

we have $\det A = 6 - 1 = 5$ and $\det B = 1 - 8 = -7$. The product is

$$AB = \begin{pmatrix} 2 + 2 & 8 + 1 \\ 1 + 6 & 4 + 3 \end{pmatrix} = \begin{pmatrix} 4 & 9 \\ 7 & 7 \end{pmatrix},$$

and $\det(AB) = 28 - 63 = -35 = 5 \cdot (-7)$.

Permutations, thirty years later

In the companion memoir of 1815, Cauchy studied how many different values a function of $n$ variables can take when the variables are permuted, continuing work by Lagrange and Ruffini. Then he dropped the subject. In 1844, back in Paris and unable to teach, he took it up again with vigour and published, within a few months, a series of papers that founded the theory of permutation groups.

That is where cycle notation appeared: $(1\;2\;3)$ means “1 goes to 2, 2 goes to 3, 3 goes to 1”. To compose two permutations, the one on the right is applied first. With $\sigma = (1\;2\;3)$ and $\tau = (1\;2)$:

$$1 \xrightarrow{\tau} 2 \xrightarrow{\sigma} 3, \qquad 2 \xrightarrow{\tau} 1 \xrightarrow{\sigma} 2, \qquad 3 \xrightarrow{\tau} 3 \xrightarrow{\sigma} 1,$$

and therefore $\sigma\tau = (1\;3)$. A set of permutations closed under composition, which Cauchy called a “system of conjugate substitutions”, is what is now called a group, the word that Galois had used in papers that would only be published in 1846.

Cauchy’s theorem

In 1845 Cauchy proved the theorem that bears his name: if a prime number $p$ divides the number of elements of a finite group, then the group has an element of order $p$, that is, some $x \ne e$ with $x^p = e$.

Cauchy’s proof was long. In 1959 James McKay found one that fits in a few lines. Consider the lists $(x_1, x_2, \ldots, x_p)$ of elements of the group $G$ with $x_1 x_2 \cdots x_p = e$. The first $p - 1$ elements are free, and the last is then determined: there are $|G|^{p-1}$ lists, a multiple of $p$. If a list has product $e$, the rotated list $(x_2, \ldots, x_p, x_1)$ does too. Under rotation, the lists group into blocks of $p$ different lists, except the constant ones $(x, x, \ldots, x)$, which stand alone, and these are exactly the ones with $x^p = e$. Since the total is a multiple of $p$ and so are the blocks, the number of constant lists is a multiple of $p$. One of them is $(e, \ldots, e)$; so there are at least $p - 1$ others, and each gives an element of order $p$.

Exercise: Cauchy's theorem in S₃

The group of permutations of $\{1, 2, 3\}$ has $6 = 2 \cdot 3$ elements. Find the elements of order 2 and of order 3 that Cauchy’s theorem guarantees.

The six elements are the identity $e$, the three swaps $(1\;2)$, $(1\;3)$, $(2\;3)$ and the two cycles $(1\;2\;3)$, $(1\;3\;2)$. Each swap, applied twice, returns to the start: order 2. Each three-cycle, applied three times, returns to the start: order 3. Check McKay’s count too: the solutions of $x^2 = e$ are $e$ and the three swaps, four in all, a multiple of 2; those of $x^3 = e$ are $e$ and the two cycles, three in all, a multiple of 3.

Exercise: the determinant of a power

Use the product theorem to compute $\det(A^{10})$ for $A = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$.

$\det A = 0 - 1 = -1$. By the product theorem, $\det(A^{10}) = (\det A)^{10} = (-1)^{10} = 1$. As a curiosity, $A^{10} = \begin{pmatrix} 89 & 55 \\ 55 & 34 \end{pmatrix}$, with Fibonacci numbers, and indeed $89 \cdot 34 - 55^2 = 3026 - 3025 = 1$.

Sources: Augustin-Louis Cauchy, “Mémoire sur les fonctions qui ne peuvent obtenir que deux valeurs égales et de signes contraires…”, Journal de l’École polytechnique (1815) · Augustin-Louis Cauchy, “Mémoire sur les arrangements que l’on peut former avec des lettres données”, Exercices d’analyse et de physique mathématique 3 (1844) · James H. McKay, “Another proof of Cauchy’s group theorem”, American Mathematical Monthly 66 (1959) · Thomas Muir, The Theory of Determinants in the Historical Order of Development, vol. 1 (1906).

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