Feit–Thompson theorem
theorem that every finite group of odd order is solvable
In mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved in the early 1960s by Walter Feit and John Griggs Thompson.
History
In the early 20th century, William Burnside conjectured that every nonabelian finite simple group has even order. Richard Brauer suggested using the centralizers of involutions of simple groups as the basis for the classification of finite simple groups, as the Brauer–Fowler theorem shows that there are only a finite number of finite simple groups with given centralizer of an involution. A group of odd order has no involutions, so to carry out Brauer's program it is first necessary to show that non-cyclic finite simple groups never have odd order. This is equivalent to showing that odd order groups are solvable, which is what Feit and Thompson proved.
The attack on Burnside's conjecture was started by Michio Suzuki, who studied CA groups; these are groups such that the centralizer of every non-trivial element is abelian. In a pioneering paper he showed that all CA groups of odd order are solvable. (He later classified all the simple CA groups, and more generally all simple groups such that the centralizer of any involution has a normal 2-Sylow subgroup, finding an overlooked family of simple groups of Lie type in the process, that are now called Suzuki groups.)
Feit, Thompson, and Marshall Hall extended Suzuki's work to the family of CN groups; these are groups such that the centralizer of every non-trivial element is nilpotent. They showed that every CN group of odd order is solvable.
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