
176 Solution by Radicals
of f . It follows easily that the map g 7→ g
0
is a group monomorphism of G into the
group S
n
of all permutations of the zeros of f . In other words, we can think of G as
a group of permutations on the zeros of f . This, in effect, was how Galois thought
of the Galois group, and for many years afterwards the only groups considered by
mathematicians were permutation groups and groups of transformations of variables.
Arthur Cayley was the first to propose a definition for an abstract group, although it
seems that the earliest satisfactory axiom system for groups was given by Leopold
Kronecker in 1870 (Huntingdon 1905).
We may restate ...