RePoSS #11: The Mathematics of Niels Henrik Abel: Continuation ...

92 Chapter 5. Towards unsolvable equations
for the substitution which transformed the permutation A1 into A2 in the above-
mentioned way. 91 He then defined ( A1) to be the product of two substitutions (A2
A6 A3 )
and ( A4 A5 ) if it gave the same result as the two applied sequentially,92 in which case
CAUCHY wrote � A1
A6
�
=
� ��
A2 A4
Furthermore, he defined the identical substitution and powers of a substitution to have
the meanings we still attribute to these concepts today. 93 The smallest integer n such
that the n th power of a substitution was the identity substitution, CAUCHY called the
degree of the substitution. 94 All these notational advances played a central part in
formalizing the manipulations on permutations and were soon generally adopted.
LAGRANGE’S Theorem. In order to demonstrate LAGRANGE’S theorem, CAUCHY
let K denote an arbitrary expression in n quantities,
A3
A5
K = K (x1, . . . , xn) .
With N = n!, he labelled the N different permutations of these n quantities
A1, . . . , AN.
The values which K would acquire when the corresponding substitutions of the form
( A1 Au ) were applied were correspondingly labelled K1, . . . , KN,
� �
A1
Ku = K for 1 ≤ u ≤ N.
Au
If these were all distinct, the expression K would obviously have N different values
when its arguments were interchanged. In the contrary case, CAUCHY assumed that
for M indices the values of K were equal
�
.
Kα = K β = Kγ = . . . .
The core of the proof was CAUCHY’S realization that if the permutation A λ was fixed
and the substitution ( Aα
A β ) was applied to A λ giving Aµ, i.e.
�
Aα
Aµ =
A β
�
A λ,
the corresponding values K λ and Kµ would be identical. Consequently, the different
values of K came in bundles of M and CAUCHY had deduced that M had to divide
n!. The central concept of degree of equivalence, which RUFFINI had introduced to mean
91 (A.-L. Cauchy, 1815a, 67).
92 (ibid., 73).
93 (ibid., 73, 74)
94 (ibid., 76). ABEL was later to change this term to the now standard order.