Burnside speculated that every non-Abelian finite simple group will have an even order. Brauer assumes that this is correct, which is the basis of the classification of finite simple groups, and proves that if the involutory centralizer is known, finite simple groups can usually be determined. Groups of odd order are not involutory groups, so to realize brower's plan, we must first prove that an acyclic finite simple group will never be of odd order. This is equivalent to proving that odd-numbered groups are solvable, which is exactly what fyt and Thompson proved. The proof of burnside's conjecture began with Naoko Suzuki, who studied the "CA group"-a group that centralizes every nonessential element Abel. In a prospective paper, he proved that all odd-order CA groups are solvable. (Then, he classified all simple CA groups, more generally, all simple groups with involutory centers of normal 2- Siroko groups, and found a rough type of Lie simple groups in the process, which is now called Suzuki group. ) fyt, Hall and Thompson extended Suzuki's results to the scope of CN groups-making the centralizer of each unnecessary element a nilpotent group. They proved that every odd-numbered CN group is solvable. Its proof is similar to Suzuki's proof, about 17 pages, which is considered as an extremely long proof in the group argument at that time. Fayette-Thompson theorem can be regarded as the next step in this process: they prove that there is no odd-order acyclic simple group in which every subgroup is solvable. This proves that every odd-numbered group is solvable, and with its smallest counterexample, there must be a simple group that can make every subgroup solvable. Although its proof is the same as the outline of CA theorem and CN theorem, its details are extremely complicated.