40 research outputs found

    Factorizations of finite groups by conjugate subgroups which are solvable or nilpotent

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    We consider factorizations of a finite group GG into conjugate subgroups, G=Ax1AxkG=A^{x_{1}}\cdots A^{x_{k}} for AGA\leq G and x1,,xkGx_{1},\ldots ,x_{k}\in G, where AA is nilpotent or solvable. First we exploit the split BNBN-pair structure of finite simple groups of Lie type to give a unified self-contained proof that every such group is a product of four or three unipotent Sylow subgroups. Then we derive an upper bound on the minimal length of a solvable conjugate factorization of a general finite group. Finally, using conjugate factorizations of a general finite solvable group by any of its Carter subgroups, we obtain an upper bound on the minimal length of a nilpotent conjugate factorization of a general finite group

    On finitely generated profinite groups II, products in quasisimple groups

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    We prove two results. (1) There is an absolute constant DD such that for any finite quasisimple group SS, given 2D arbitrary automorphisms of SS, every element of SS is equal to a product of DD `twisted commutators' defined by the given automorphisms. (2) Given a natural number qq, there exist C=C(q)C=C(q) and M=M(q)M=M(q) such that: if SS is a finite quasisimple group with S/Z(S)>C| S/\mathrm{Z}(S)| >C, βj\beta_{j} (j=1,...,M) (j=1,...,M) are any automorphisms of SS, and qjq_{j} (j=1,...,M) (j=1,...,M) are any divisors of qq, then there exist inner automorphisms αj\alpha_{j} of SS such that S=1M[S,(αjβj)qj]S=\prod_{1}^{M}[S,(\alpha_{j}\beta_{j})^{q_{j}}]. These results, which rely on the Classification of finite simple groups, are needed to complete the proofs of the main theorems of Part I.Comment: 34 page

    Gauss decomposition for Chevalley groups, revisited

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    In the 1960's Noboru Iwahori and Hideya Matsumoto, Eiichi Abe and Kazuo Suzuki, and Michael Stein discovered that Chevalley groups G=G(Φ,R)G=G(\Phi,R) over a semilocal ring admit remarkable Gauss decomposition G=TUUUG=TUU^-U, where T=T(Φ,R)T=T(\Phi,R) is a split maximal torus, whereas U=U(Φ,R)U=U(\Phi,R) and U=U(Φ,R)U^-=U^-(\Phi,R) are unipotent radicals of two opposite Borel subgroups B=B(Φ,R)B=B(\Phi,R) and B=B(Φ,R)B^-=B^-(\Phi,R) containing TT. It follows from the classical work of Hyman Bass and Michael Stein that for classical groups Gauss decomposition holds under weaker assumptions such as \sr(R)=1 or \asr(R)=1. Later the second author noticed that condition \sr(R)=1 is necessary for Gauss decomposition. Here, we show that a slight variation of Tavgen's rank reduction theorem implies that for the elementary group E(Φ,R)E(\Phi,R) condition \sr(R)=1 is also sufficient for Gauss decomposition. In other words, E=HUUUE=HUU^-U, where H=H(Φ,R)=TEH=H(\Phi,R)=T\cap E. This surprising result shows that stronger conditions on the ground ring, such as being semi-local, \asr(R)=1, \sr(R,\Lambda)=1, etc., were only needed to guarantee that for simply connected groups G=EG=E, rather than to verify the Gauss decomposition itself

    Minimal length factorizations of finite simple groups of Lie type by unipotent Sylow subgroups

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    We prove that every finite simple group G of Lie type satisfies G = UU−UU − where U is a unipotent Sylow subgroup of G and U − is its opposite. We also characterize the cases for which G = UU−U. These results are best possible in terms of the number of conjugates of U in the above factorizations

    Strong approximation methods in group theory, an LMS/EPSRC Short course lecture notes

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    These are the lecture notes for the LMS/EPSRC short course on strong approximation methods in linear groups organized by Dan Segal in Oxford in September 2007.Comment: v4: Corollary 6.2 corrected, added a few small remark

    Commutator width in Chevalley groups

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    The present paper is the [slightly expanded] text of our talk at the Conference “Advances in Group Theory and Applications” at Porto Cesareo in June 2011. Our main results assert that [elementary] Chevalley groups very rarely have finite commutator width. The reason is that they have very few commutators, in fact, commutators have finite width in elementary generators. We discuss also the background, bounded elementary generation, methods of proof, relative analogues of these results, some positive results, and possible generalisations
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