
166 Solubility and Simplicity
and if u = (145)N contains
u
−1
(t
−1
xtx
−1
)u = (45)(23)
so that N contains
(45)(23)(14)(23) = (145)
contradicting the assumption that every element of N is a product of disjoint 2-cycles.
Hence A
n
is simple if n ≥ 5.
In fact A
5
is the smallest non-abelian simple group. This result is often attributed
to Galois, but Neumann (2011), in his translation of Galois’s mathematical writings,
points out on pages 384–385 that alternating groups are not mentioned in any sig-
nificant work by Galois, and that the methods available to him were inadequate to
eliminate various orders for a potential simple group, such as 56. Although it seems ...