Abstract
Let Φ be a root system of typeA ℓ, ℓ ≧ 2,D ℓ, ℓ ≧ 4 orE ℓ, 6 ≧ ℓ ≧ 8 andG a group generated by nonidentity abelian subgroupsA r,r∈Φ, satisfying:
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(i)
[A r, As]=1 ifs≠−r and ∉ Φ,
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(ii)
[A r, As]≦A r+s ifr+s∈Φ,
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(iii)
X r=〈Ar, A−r〉 is a rank one group.
Then it is shown, using [3], thatG is a central product of Lie-type groups corresponding to a decomposition of Φ into root-subsystems.
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References
R. Carter, Simple groups of Lie-type. Pure Appl. Math. XXVIII, London 1972.
F. G. Timmesfeld, Abstract root subgroups and simple groups of Lie-Type. Monograph. Math.95, Basel 2001.
F. G. Timmesfeld, Presentations for certain Chevalley Groups. Geom. Ded.73, 85–117 (1998).