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

86 Chapter 5. Towards unsolvable equations
simple permutations
composite permutations
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⎧
⎨
⎩
powers of a cycle
powers of a non-cycle
intransitive ones
transitive, imprimitive ones
transitive, primitive ones
Table 5.1: RUFFINI’s classification of permutations
systems of conjugate substitutions. 77 Thus, a permutation for RUFFINI corresponded to a
collection of interchangements (i.e. transitions from one arrangement of symbols e.g.
123 . . . n to another e.g. 213 . . . n) which left the given function formally unaltered. 78
Classification of permutations. RUFFINI divided his permutations into simple ones
which were generated by iterations (i.e. powers) of a single interchangement 79 and
composite ones generated by more than one interchangement. His simple permutations
consisting of powers of a single interchangement were subdivided into two types dis-
tinguishing the case in which the single interchangement consisted of a single cycle
from the case in which it was the product of more than one cycle.
RUFFINI’S composite permutations were subsequently subdivided into three types. 80
A permutation (i.e. set of interchangements) was said to be of the first type if two
arrangements existed which were not related by an interchangement from the permu-
tation. 81 Translated into the modern terminology of permutation groups, this type
corresponds to intransitive groups. RUFFINI defined the second type to contain all per-
mutations which did not belong to the first type and for which there existed some
non-trivial subset of roots S such that, in modern notation, σ (S) = S or σ (S) ∩ S = ∅
for any interchangement σ belonging to the permutation. Such transitive groups were
later termed imprimitive. The last type consisted of any permutation not belonging to
any of the previous types, and thus corresponds to primitive groups.
Building on this classification of all permutations into the five types (table 5.1),
RUFFINI introduced his other key concept of degree of equivalence (Italian: “grado di
uguaglianza”) of a given function f of the n roots of an equation as the number of dif-
ferent permutations not altering the formal value of f . Denoting the degree of equiva-
lence by p, RUFFINI stated the result of LAGRANGE (see section 5.2) that p must divide
n!.
77 (Burkhardt, 1892, 133).
78 In modern notation: With f the given function of n quantities, a permutazione to RUFFINI was a set
G ⊆ Σn such that f ◦ σ = f for all σ ∈ G.
79 RUFFINI’S simple permutations correspond to the modern concept of cyclic permutation groups.
80 (Ruffini, 1799, 163).
81 I.e. there exists two arrangements a and b such that σ (a) �= b for all σ in the set of interchangements.