On the theory of solvable linear groups

Abstract

| 1 . D e s i g n a t i o n s . S u m m a r y o f K n o w n R e s u l t s Fundamental in the theory, of solvable l inear groups is the case of p r imi t ive solvable l inea r groups. .Many impor tant p rope r t i e s of solvable groups of ma t r i ce s a r e deduced f rom the p roper t i e s :o f p r imi t ive so lvable groups . In pa r t i cu la r , the c lass i f ica t ion of the max imum solvable subgroups of a comple te l inear group, o v e r a field reduces by and la rge t o the c lass i f ica t ion of the m a x i m u m p r imi t i ve so lvable subgroups {|6], Chap. 1, I i ) . In the p resen t a r t i c le we invest igate only p r imi t ive solvable l inear groups. We adhe re to the following notation: A is.an a r b i t r a r y field, f/ is ,an a lgebra ica l ly -c losed field. G is a p r i m i t i v e I r reduc ib le maximum solvable subgroup of group GL (n, A), F i s the max imum Abelian normal ~ v i s o r of G, V is the cen t ra l i ze r of F in G, and A is a subgroup of G such that {i) A / F is an Abelian normal d iv i sor of the group G/F ; (ii) A / F c V /F ; {iii) A / F is the max imum of the subgroups of G/F that have p r o p e r t i e s (i) and (ti). We se t 0 Et l = i E l 0 ]" We define the s implec t i c group $9(2/, &) as the group of all m a t r i c e s h f rom GL(2/, A) that sa t is fy the condition t h f t h = q~l, where th is the t r anspose of h. If A= GF(p), then in place of Sp{2L &) we wri te ~ ( 2 / . p). A subgroup H of the group Sp(2L A) is called s r educ ib le if there is =~n integer v, 1 z: v<_l, and a ma t r ix b E Sp(2l, A) such that all m a t r i c e s of the group b-tHB have the fo#m