The Feit-Thompson Theorem states that every finite group of odd order is solvable. This statement is clearly equivalent to the following: there is no non-abelian simple group of odd order. The theorem, first conjectured by Burnside, was proved in 1963 by W. Feit and J.G. Thompson in [FT]. Two papers, which prove the theorem in special cases, preceded the appearance of [FT]. In [Su], M. Suzuki proved the theorem for CA-groups of odd order: a group G is a CA-group if, for every element x ≠ 1 of G, CG (x) is abelian. In [FHT], the theorem was shown for the CN-groups of odd order: a group G is a CN-group if, for every element x ≠ 1 of G, CG(x) is nilpotent.

Each of these proofs is divided into two parts. In the first part, a minimal counterexample G to the theorem is considered and the structure of the maximal subgroups of G is studied. This part is very short in [Su], but is much more complicated in [FHT], and considerably more so in [FT]. In the second part, a contradiction is obtained by the use of character theory. The existence of isometries between virtual characters of maximal subgroups of G and virtual characters of G is one of the basic tools. In [FT], this second part leaves a residual case in which no contradiction arises. This case is eliminated in the final chapter of [FT], by explicit calculations with relations between elements of G.