Thanks to Tony Mann who contacted Peter Neumann my attention was called to
Peter Neumann's review of Camille Jordan's "Traite\" in Math. Reviews. There I
got the reference to a paper of Wieland and Huppert and they led me to
Camille Jordan, Sur la limite de transitivite/ des groupes non alterne/s.
Bull. Soc. math. de France 1, 40--71, 1873
On page 70/71 he rectifies the theorem on the transitivity of normal subgroups
of n-fold transitive groups saying and proving that the only exceptions are
triply transitive groups with a regular normal subgroup G the order of which
is a power of 2, the group G being, of course, elementary abelian.
However, he does not say, how this does affect his proof of the simplicity of
the alternating group of degree k.
In the above paper, page 41, he still says that the alternating group on k
letters is (k - 1)-fold transitive.
A remark to the posting of John Pais concerning Sylow p-subgroups. The
existence of such groups in the case of the symmetric groups have been
established already by Cauchy:
Me/moire sur les arrangements que l'on peut former avec des lettres donne/es
et sur les permutations ou substitutions a\ l'aide desquelles on passe d'un
arrangement a\ un autre. In: Exercise d'analyse et de physique mathe/matique.
Vol. III, Paris 1844. Oeuvres 2. Ser. Vol. XIII, Paris 1932, p. 171--282
In this memoir, one finds also his famous theorem that a subgroup of the
symmetric group contains an element of order p, provided, p is a prime dividing
the order of the subgroup. His proof uses doubler-coset-techniques. The same
techniques may also be used to yield Sylow's theorems.
Best regards, Heinz Lueneburg