GENERATION OF RELATIVE COMMUTATOR SUBGROUPS IN CHEVALLEY GROUPS

Let Phi be a reduced irreducible root system of rank greater than or equal to 2, let R be a commutative ring and let I, J be two ideals of R. In the present paper we describe generators of the commutator groups of relative elementary subgroups [E(Phi, R, I), E(Phi, R, J)] both as normal subgroups of the elementary Chevalley group E(Phi, R), and as groups. Namely, let x(alpha)(xi), alpha is an element of Phi, xi is an element of R, be an elementary generator of E(Phi, R). As a normal subgroup of the absolute elementary group E(Phi, R), the relative elementary subgroup is generated by x(alpha)(xi), alpha is an element of Phi, xi is an element of I. Classical results due to Stein, Tits and Vaserstein assert that as a group E(Phi, R, I) is generated by z(alpha)(xi, eta), where alpha is an element of Phi, xi is an element of I, eta is an element of R. In the present paper, we prove the following birelative analogues of these results. As a normal subgroup of E(F, R) the relative commutator subgroup [E(Phi, R, I), E(Phi, R, J)] is generated by the following three types of generators: (i) [x(alpha)(xi), z(alpha)(zeta, eta)], (ii) [x(alpha)(xi), x(-alpha)(zeta)] and (iii) x(alpha)(xi zeta), where alpha is an element of Phi, xi is an element of I, zeta is an element of J, eta is an element of R. As a group, the generators are essentially the same, only that type (iii) should be enlarged to (iv) z(alpha)(xi zeta, eta). For classical groups, these results, with many more computational proofs, were established in previous papers by the authors. There is already an amazing application of these results in the recent work of Stepanov on relative commutator width.