可解完全线性群的阶

The Orders of Solvable Completely Linear Groups

证明了可解群阶的定理:令V≠0是有限拟本原G 模,|V|=qn,素数q>0,G为完全线性群GL(V)的一可解完全可约子群,则a)|G|≤|V|α/λ,b)若2 |G|,且q≠2,则|G|≤|V|3/2/241/3。除非qn=23、26、28、29、32、34或54,或者G≌(Γ(22)wrZ2)F、Γ(24)F或(S3wrZ2)F,F是25阶超特殊群。常数4α=6(3)1/2,即1.68<α<1.69,且常数λ=31/2。

The author main shows that a theorem about the orders of soluble groups;Let V≠0 be a faithful.completely reducible,quasi-primitive and finite G-module for a solvable group G.Set char|V|=q>0.a):|G|≤|V|~α/λ.b):If 2|G|,and q≠2,then |G|≤|V|^(3/2)/24^(1/3).Unless q^n=2~3,2~6,2~8,2~9,3~2,3~4 or 5~4,or G≌(Γ(2~2)wrZ_2)F,Γ(2~4)F or (S_3wrZ_2)F.Where F isextra-special group of order 2~5.4~α=6.3^(1/2),i.e.1.68<α<1.69,and λ=3^(1/2).