Feit-Thompson theorem

The set of Feit-Thompson , named after Walter Feit and John G. Thompson , is a set of the mathematical branch of group theory .

Formulation of the sentence

• Every group with an odd group order can be resolved .

The English-language literature also speaks of the odd-order theorem .

Notes on Evidence

Despite the impressively simple formulation of this sentence, no available evidence is known. The theorem was already suspected by William Burnside in 1911 , but could not be proven until 1963 by W. Feit and JG Thompson. The original proof is more than 250 pages, fills the complete number 3 of volume 13 of the Pacific Journal of Mathematics and is freely available.

In the period that followed, H. Bender and G. Glauberman in particular made some simplifications, but no breakthrough has been achieved in terms of the length of evidence and the original evidence structure has remained essentially unchanged. They left out the character-theoretical part of the proof, but it was simplified by Thomas Peterfalvi. A description of the evidence can be found in D. Gorenstein's textbook given below .

Georges Gonthier and colleagues succeeded in verifying the evidence with Coq after six years of work in 2012 .

meaning

Finite easy groups

If a finite group is simple and not cyclic of prime order, the group order is even. This follows immediately from Feit-Thompson's theorem, because an odd group has a non-trivial normal divisor as a solvable group or is cyclic in the prime number order. Since among the Abelian groups according to the main theorem about finitely generated Abelian groups, the simple groups are exactly the cyclic groups of prime order, this can be reformulated as follows:

• Non-Abelian simple groups have just group order.

This statement is equivalent to Feit-Thompson's theorem, because if a group cannot be resolved, then every composition series contains a non-Abelian simple group and, according to the assumption, its order is even and, of course, divisor of the group order, which is therefore also straight.

So every finite simple nonabelian group has an even group order and thus contains an element of order 2, a so-called involution, according to Cauchy's theorem . The study of the centralizers of such involutions is the starting point for the classification of finite simple groups .

Schur-Zassenhaus's theorem

According to Schur-Zassenhaus's theorem , in a finite group with normal divisors , so that the orders of and of the factor group are relatively prime , there is a subgroup with and . ${\displaystyle G}$ ${\displaystyle N}$ ${\displaystyle N}$ ${\displaystyle G/N}$ ${\displaystyle U}$ ${\displaystyle G=NU}$ ${\displaystyle N\cap U=\{1\}}$

Such subgroups are called a complement to . The following unambiguity statement was traditionally proven: ${\displaystyle U}$ ${\displaystyle N}$

• If additional or can be resolved, two complements are to be conjugated . ${\displaystyle N}$ ${\displaystyle G/N}$ ${\displaystyle N}$

With Feit-Thompson's theorem, the additional solvability requirement can be dispensed with, because if and have coprime group orders, one of these two group orders must be odd. ${\displaystyle N}$ ${\displaystyle G/N}$

Individual evidence

1. ^ W. Feit, JG Thompson, Solvability of groups of odd order , Pacific Journal of Mathematics, Volume 13, 1963, pages 775-1029
2. ↑ Original article as pdf
3. ^ H. Bender, G. Glauberman: Local analysis for the odd order theorem , Cambridge University Press, London Mathematical Society Lecture Note Series (1994), Volume 188, ISBN 978-0-521-45716-3
4. ↑ Thomas Peterfalvi: Character theory for the odd order theorem , London Mathematical Society Lecture Note Series 272, Cambridge University Press, 2000
5. ^ D. Gorenstein: Finite Groups , AMS Chelsea Publishing (1980), 2nd edition, ISBN 978-0-82184342-0
6. ^ Feit-Thompson proved in Coq , Microsoft Research-Inria, September 20, 2012, Web Archive
7. ↑ H. Kurzweil, B. Stellmacher: Theory of finite groups , Springer-Verlag (1998), ISBN 3-540-60331-X , Section 6.2.1