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A root system for the Lyons group. (English) Zbl 0642.20015

Recently the question came up how the geometric properties of the Chevalley groups (root systems, root groups, etc.) may be transferred to sporadic groups.
We construct a system of 36 root groups for the Lyons group Ly corresponding to the lines in an apartment of Kantor’s Tits geometry. Such an apartment is a two-dimensional simplicial complex isomorphic to a torus. Its geometric structure leads to relations between the generators (“roots”) of the root groups which define a group \(\Gamma\). We show that Ly is a factor group of \(\Gamma\). Furthermore, there is strong evidence that \(\Gamma\) \(\cong Ly\). Many large subgroups of Ly (among them, e.g., \(G_ 2(5)\), \(5^{1+4}:4S_ 6\), 5 \(3.SL_ 3(5)\); \(2^{\wedge}A_{11}\), \(3^{\wedge}U_ 4(3)\), \(3^{\wedge}Mc:2)\) can be constructed and identified in a natural way with the help of our root system.
Reviewer: W.Neutsch

MSC:

20D08 Simple groups: sporadic groups
51D20 Combinatorial geometries and geometric closure systems
20F65 Geometric group theory
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References:

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