40 research outputs found
Factorizations of finite groups by conjugate subgroups which are solvable or nilpotent
We consider factorizations of a finite group into conjugate subgroups,
for and ,
where is nilpotent or solvable. First we exploit the split -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
We prove two results. (1) There is an absolute constant such that for any
finite quasisimple group , given 2D arbitrary automorphisms of , every
element of is equal to a product of `twisted commutators' defined by
the given automorphisms.
(2) Given a natural number , there exist and such that:
if is a finite quasisimple group with ,
are any automorphisms of , and are any
divisors of , then there exist inner automorphisms of such
that .
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
In the 1960's Noboru Iwahori and Hideya Matsumoto, Eiichi Abe and Kazuo
Suzuki, and Michael Stein discovered that Chevalley groups over a
semilocal ring admit remarkable Gauss decomposition , where
is a split maximal torus, whereas and
are unipotent radicals of two opposite Borel subgroups
and containing . 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 condition
\sr(R)=1 is also sufficient for Gauss decomposition. In other words,
, where . 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 , rather than to verify the Gauss decomposition itself
Minimal length factorizations of finite simple groups of Lie type by unipotent Sylow subgroups
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
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
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