146 research outputs found
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
Algebraic properties of profinite groups
Recently there has been a lot of research and progress in profinite groups.
We survey some of the new results and discuss open problems. A central theme is
decompositions of finite groups into bounded products of subsets of various
kinds which give rise to algebraic properties of topological groups.Comment: This version has some references update
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