tailieunhanh - Đề tài " On finitely generated profinite groups, II: products in quasisimple groups "

We prove two results. (1) There is an absolute constant D such that for any finite quasisimple group S, given 2D arbitrary automorphisms of S, every element of S is equal to a product of D ‘twisted commutators’ defined by the given automorphisms. (2) Given a natural number q, there exist C = C(q) and M = M (q) such that: if S is a finite quasisimple group with |S/Z(S)| C, βj (j = 1, . . . , M ) are any automorphisms of S, and qj (j = 1, . . . , M ) are. | Annals of Mathematics On finitely generated profinite groups II products in quasisimple groups By Nikolay Nikolov and Dan Segal Annals of Mathematics 165 2007 239-273 On finitely generated profinite groups II products in quasisimple groups By NiKOLAy Nikolov and Dan Segal Abstract We prove two results. 1 There is an absolute constant D such that for any finite quasisimple group S given 2D arbitrary automorphisms of S every element of S is equal to a product of D twisted commutators defined by the given automorphisms. 2 Given a natural number q there exist C C q and M M q such that if S is a finite quasisimple group with S Z S C 3j j 1 . M are any automorphisms of S and qj j 1 . M are any divisors of q then there exist inner automorphisms aj of S such that S nM S aj 3j q . These results which rely on the classification of finite simple groups are needed to complete the proofs of the main theorems of Part I. 1. Introduction The main theorems of Part I NS were reduced to two results about finite quasisimple groups. These will be proved here. A group S is said to be quasisimple if S S S and S Z S is simple where Z S denotes the centre of S. For automorphisms a and 3 of S we write Ta 0 x y x 1y 1xay3. Theorem . There is an absolute constant D E N such that if S is a finite quasisimple group and a1 31 . aD 3d are any automorphisms of S then S Tai 0l S S .T S S . Theorem . Let q be a natural number. There exist natural numbers C C q and M M q such that if S is a finite quasisimple group with S Z S C 3d. 3m are any automorphisms of S and q1 . qM are Work done while the first author held a Golda-Meir Fellowship at the Hebrew University of Jerusalem. 240 NIKOLAY NIKOLOV AND DAN SEGAL any divisors of q then there exist inner automorphisms a1 . aM of S such that S S 1A qi . S aMPm qM . These results are stated as Theorems and in the introduction of NS . Both may be seen as generalizations of Wilson s theorem W that every element of any finite simple group is .