A rank 3 geometry for the sporadic simple group of Suzuki - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T11:17:00Z http://mathoverflow.net/feeds/question/59410 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/59410/a-rank-3-geometry-for-the-sporadic-simple-group-of-suzuki A rank 3 geometry for the sporadic simple group of Suzuki Thomas Connor 2011-03-24T09:42:02Z 2011-04-10T13:28:59Z <p>Dear everyone,</p> <p>I am actually studying coset geometries (in the sense of Tits and Buekenhout) for the sporadic simple group of Suzuki. I came aware that Buekenhout found in 1979 a geometry over the following diagram</p> <pre><code> c 6 O----------O----------O 1 4 4 </code></pre> <p>However, I couldn't find any information about the maximal (or minimal) parabolic subgroups of this geometry.</p> <p>Has anyone ever studied this geometry? Is there a paper where I could find the informations I am looking for?</p> <p>As usual, thanks in advance!</p> http://mathoverflow.net/questions/59410/a-rank-3-geometry-for-the-sporadic-simple-group-of-suzuki/59676#59676 Answer by Thomas Connor for A rank 3 geometry for the sporadic simple group of Suzuki Thomas Connor 2011-03-26T19:43:31Z 2011-03-26T19:43:31Z <p>I finally found the maximal parabolic subgroups of this geometry. Let us first denote the types of the elements with 0,1 and 2 when reading the diagram from left to right, and let us denote with $G_0$, $G_1$ and $G_2$ the stabilizer of an element of type 0, 1 and 2 respectively. Then we have:</p> <p>$$G_0 = G_2(4), G_1 = 2^{2+8}:(A_5 \times S_3), G_2 = 2^{4+6} : 3 A_6$$ which are all maximal subgroups of $Sz$, and the Borel is $$B = 2^{12}.3^2$$ Historically, I read that this geometry was built using polar spaces (see Francis Buekenhout, Diagrams for geometries and groups, Journal of Combinatorial Theory A, 27, 121-151, 1979). However, I have not studied yet how to build it geometrically.</p>