- THEORY OF EQUATIONS 445
Ruffini had shown that it was not possible to exhibit a function of five variables
that could be changed into three different functions or four different functions by
permuting the variables. It was this work that Cauchy proposed to generalize.
Cauchy's theorem was an elegant piece of work in the theory of finite permu-
tation groups. To prove it, he had to invent a good deal of that theory. He pointed
out that the number of permutations Í equals n!, and that the number of those
permutations that leave the function unchanged is a divisor of JV, which he de-
noted M. In a manner now familiar, he showed that the number of different values
(that is, different functions of the variables) that can be obtained by permuting
the variables is R = N/M, and that if S is a permutation that leaves the function
unchanged and Ô changes its value from Ê to K', then ST also changes its value
from Ê to K'. He then introduced cyclic permutations and what we now call the
order of a cyclic permutation. The matrix notation now sometimes used for per-
mutations and the notation (áâç) for a permutation that maps á to â, â to 7, and
7 to a, leaving all other elements fixed, was introduced in this paper.
Cauchy showed that if a permutation U is of order m, the complete set of
permutations breaks up into N/m pairwise disjoint subsets (now called cosets) of
m elements each. If m > R, which means Ì > N/m, some coset must contain two
distinct elements S and Ô that leave the function invariant. When m is a prime p,
this fact implies that some power Us with s between 1 and ñ — 1 leaves the function
invariant, and since every power Usk then leaves the function invariant, it follows
that all powers of U leave the function invariant. If ñ is 2, this is not a strong
statement, since R = 1 in that case, and all permutations whatsoever leave the
function invariant. For ñ > 2, it implies that the set of permutations that leave the
function invariant contains all permutations of order p.
Cauchy then showed that this set must contain all permutations of order 3,
by explicitly writing any permutation of order 3 as the composition of two permu-
tations of order p.^6 It then followed that the permutation group can produce at
most two different functions. For this case Cauchy showed that the function must
be of the form Ê + SV, where Ê and S are symmetric and V is the Vandermonde
determinant mentioned above, which switches sign when any two of its arguments
are interchanged.
Besides the notation for permutations and cycles, Cauchy also invented some
of the terminology of group theory, including the word index (indice) still used for
the number of cosets of a subgroup of a finite group. For the number of elements
Ì in the subgroup, he used the term indicial (or indicative) divisor (diviseur in-
dicatif). He proposed the name substitution (of one permutation into another) for
the composition of two permutations, and he called two permutations equivalent if
they produce the same function, that is, they are equal modulo the subgroup of
permutations that leave the function invariant. To picture cyclic permutations of
finite order, he suggested arranging the distinct powers as the vertices of a regular
polygon and thinking of the composition of two of them as a clockwise rotation (he
said "a rotation from east to west") of the polygon. Such an arrangement suggests
studying the symmetries of these polygons. However, although he frequently re-
ferred to "groups of indices" in this paper, he did not define the notion of a group
in its modern sense.
The number Í = ç! has no prime factors larger than n, so that ñ < ç in any case.