Structures And Maps Preserving Commutators Of Classical Groups Over Rings

The structure of classical groups over rings is an important subject of algebraic K1-theory and of classical groups. This thesis mainly deals with ‘sandwich theorem’ ofquadratic groups over commutative rings and ‘generalized sandwich theorem’ of quadrat-ic groups over Banach algebras by form ideals. Finally, map preserving commutators, ageneralized automorphism, of unitary groups over fields is described. The main results ofthe paper are summarized as follows.In the middle of1990s, the structure of subgroups of quadratic groups which arenormalized by elementary subgroups over commutative rings (i.e.,‘sandwich theorem’)had been described. But the level ideals used in the study were usual ones, rather thanform ideals, that is, the result obtained before did not concern the form parameters, whichis indeed now called ‘weak sandwich theorem’. In this thesis, we studied the structureof subgroups of quadratic groups normalized by elementary subgroups over commutativerings by form ideals, and improved ‘sandwich theorem’ of quadratic groups over com-mutative rings. Meanwhile, we gave a direct proof of ‘weak sandwich theorem’ withoutusing localization method.The classification of the subnormal subgroups of classical groups is related to thedescription of the subgroups of classical groups normalized by relative elementary sub-groups. In the beginning of1980s, Anthony Bak proposed a conjecture concerning the de-scription of subgroups of quadratic groups normalized by relative elementary subgroupsover commutative rings, which is also called ‘generalized sandwich theorem’. This the-sis positively answered Bak’s conjecture for Banach algebras. In the study we obtaineda complete description of subnormal structure of quadratic groups over Banach algebrasfor n≥4.A bijective map on a group G is called a map preserving commutators of G, writtenas PC-map, if preserves commutators, that is,([x, y])=[(x),(y)](x, y∈G). It isa generalization of automorphism of groups. PC-maps of the solvable group Tn(F) over afield F had been proven as that each PC-map of Tn(F) is a product of some automorphismsand so-called central PC-map of Tn(F). Based on these, this thesis studied PC-maps of thestandard Borel subgroup B2n(F) of unitary groups over a field F whose character not2and cardinality bigger than9. By commutator operations and matrix computing technique, wecompletely depict all PC-maps on B2n(F) for n≥3.