> Dear Colleagues,
>
> I just began studying Camille Jordan's Trait'e. The day before yesterday, I
> saw that Jordan's proof of the simplicity of the alternating group of degree
> at least five is wrong because it is based on a theorem which is wrong.
<snip>
> I am astounded that Jordan did not see that the alternating group on n letters
> cannot be (n - 1)-fold transitive.
>
> Here are my questions. Has anybody ever seen it mentioned that Jordan's
> attempted proof of the simplicity of the alternating group failed?
Here is a little information that I found in The Theory of Groups, Marshall Hall,
Jr., Macmillan, 1959. On p. 72, Hall says:
" In 1872 Jordan [2] showed that a finite quadruply transitive group in which only
the identity fixes four letters must be one of the following groups: the symmetric
group on four or five letters; the alternating group on six letters, or the Mathieu
group on eleven letters.
Jordan's theorem on quadruply transitive groups is generalized here in two ways.
..."
The citation [2] given by Hall is Jordan's: Recherches sur les substitutions, J.
Math. Pures Appl. (2) 17 (1872), 351-363. So, if there is some infelicity in the
Trait'e (1870) regarding transitivity, then apparently by 1872 Jordan must have had
it properly sorted out.
> Who gave the
> first correct proof of this fact?
>
Yes, I would like to know this too. All my amateur sleuthing could determine is that
Galois seems to be credited with the proof of the simplicity of A_5. Among others, I
found an interesting (though without references) remark in the book: P. Neumann, G.
Stoy, and E. Thompson, Groups and Geometry, Oxford, 1994, p. 95:
"Recognition of the importance of simplicity is one of the contributions that Galois
made to mathematics. In a famous letter to his friend Auguste Chevalier, written
during the night before his fatal duel in 1832, he defines what he means by a simple
group and goes on to assert that the smallest simple group that is not cyclic of
prime order has order 5.4.3, that is, 60. We cannot now be certain how Galois knew
this for, although the statement can be verified quite easily usin Sylow's theorems
... it is hard to see how one could have proceeded in Galois' day, forty years before
Sylow's paper was published. (There is, however, some evidence to suggest that Galois
might at one time have known or guessed a significant part of the Sylow theorems.)
..."
Best wishes from mild and sunny St. Louis,
John Pais