William Burnside (1911, p. 503 note M) conjectured that every nonabelian finite simple group has even order.
This is equivalent to showing that odd order groups are solvable, which is what Feit and Thompson proved.
The Feit–Thompson theorem can be thought of as the next step in this process: they show that there is no non-cyclic simple group of odd order such that every proper subgroup is solvable.
Many of the techniques they introduced in their proof, especially the idea of local analysis, were developed further into tools used in the classification.
Perhaps the most revolutionary aspect of the proof was its length: before the Feit–Thompson paper, few arguments in group theory were more than a few pages long and most could be read in a day.
We let G be a non-abelian (minimal) simple group of odd order satisfying the CA condition.
The normalizers of these maximal abelian subgroups turn out to be exactly the maximal proper subgroups of G. These normalizers are Frobenius groups whose character theory is reasonably transparent, and well-suited to manipulations involving character induction.
The proof of the CN-case is already considerably more difficult than the CA-case: the main extra problem is to prove that two different Sylow subgroups intersect in the identity.
A key step is the proof of the Thompson uniqueness theorem, stating that abelian subgroups of normal rank at least 3 are contained in a unique maximal subgroup, which means that the primes p for which the Sylow p-subgroups have normal rank at most 2 need to be considered separately.
Roughly speaking, this theory says that the Dade isometry can be extended unless the groups involved have a certain precise structure.
By step 2, we have a complete and precise description of the character table of the CA group G. From this, and using the fact that G has odd order, sufficient information is available to obtain estimates for |G| and arrive at a contradiction to the assumption that G is simple.
More character-theoretic arguments show that they cannot be of types IV or V. The two subgroups have a precise structure: the subgroup S is of order pq×q×(pq–1)/(p–1) and consists of all automorphisms of the underlying set of the finite field of order pq of the form x→axσ+b where a has norm 1 and σ is an automorphism of the finite field, where p and q are distinct primes.
(In particular, the first number divides the second, so if the Feit–Thompson conjecture is true, it would assert that this cannot happen, and this could be used to finish the proof at this point.
To eliminate this final case, Thompson used some fearsomely complicated manipulations with generators and relations, which were later simplified by Peterfalvi (1984), whose argument is reproduced in (Bender & Glauberman 1994).
The proof examines the set of elements a in the finite field of order pq such that a and 2–a both have norm 1.
Then a rather difficult argument using generators and relations in the group G shows that the set is closed under taking inverses.