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. The
theorem Jordan claims to be true is the following:
If N is a normal subgroup of an n-fold transitive group, then N is (n - 1)-fold
transitive.
He draws from this theorem the conclusion that the alternating group on n
letters is (n - 1)-fold transitive and a normal subgroup of it is (n - 2)-fold
transitive. The above theorem is, of course, false, as already the alternating
group on n letters shows which is sharply (n - 2)-transitive. Moreover, given
a vector space V over GF(2) then any two non-zero vectors are linearly
independent. Hence the general linear group GL(V) on this vector space acts
2-transitively on the set of non-zero vectors. Thus the group of all mappings
x --> g(x) + a
with g in GL(V) and a in V acts triply transitively on V. But the normal
subgroup of the mappings x --> x + a acts sharply 1-transitively on V. If
the dimension of V is 4 then GL(V) is isomorphic to the alternating group of
degree 8. It contains the alternating group on 7 letters acting 2-transitively
on the set of non-zero vectors of $V$ as well. Again, one gets a 1- but not
2-transitive normal subgroup of a triply transitive group. A most beautiful
proof of the fact that A_7 acts 2-transitively on the non-zero vectors of V in
E. H. Moore, Concerning the general equations of the seventh and eighth
degrees. Math. Annalen 51, 417-444, 1899.
Burnside, in the second edition of his "Theory of groups of finite order" from
1911 (I consulted the Dover reprint), gives a proof which is very much in the
spirit of Jordan. But Burnside shows directly that N contains a 3-cycle. He does
not mention Jordan. Dickson in his "Linear groups" takes the simplicity of the
alternating groups for granted. There seems to be no quotation in this context.
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? Who gave the
first correct proof of this fact?
Best regards, Heinz Lueneburg