On the Normal Subgroups of $G_2(A)$

Abstract

We give a characterization theorem for the E(A)-normalized subgroups of G2 (A), where A is any commutative ring. This is the last of the simple Chevalley-Demazure group-schemes for which such a theorem is lacking. INTRODUCTION In this paper we prove a theorem (3.6) which characterizes the E(A)-normalized subgroups of G2(A) for an arbitrary commutative ring A. The theorem brings to completion the larger project of finding characterization theorems for E(A)normalized subgroups of simple Chevalley-Demazure group-schemes G(A) of rank > 2 over arbitrary commutative rings. The bulk of t;he project was done by Vaserstein in his essential paper [16]. There he proved the standard characterization theorem, given by a ladder corresponding to an ideal, for all commutative rings for all G of rank > 2 except those of types B2= C2 and G2. Vaserstein also suggested in that paper that previous work by Abe [1] and Abe-Suzuki [3] could be improved to yield more. This was accomplished by Abe in [2], wherein he showed that E(A)normalized subgroups of G(A) could be characterized by ladders determined by pairs of ideals, with the exception that for G of type B2 = C2 or G2 rings A having residue class fields of two elements must be excluded. Our paper [8] subsequently succeeded in removing this restriction for G of type C2 (and so also for G of type B2) by recognizing that the presence of residue fields of two elements required lowering the bottom of the ladder. This left only the case G of type G2 to be finished; the present paper does so by applying the lower bottom rung idea of [8] (see also [7]) to this case. We now give a more explicit description of our results. Let A be an arbitrary commutative ring and denote by G2(A) the group of type G2 defined over A. The main theorem (3.6) is that a subgroup N of G2(A) is E(A)-normalized if and only if C(J, J') C N C G(J, J') for a pair of ideals J, J' satisfying the condition J3 C J' C J. Here, J3 denotes the ideal of A generated by all elements of the form x3 and 3x for all x in J, so that (J, J') is an admissible pair in the sense of Abe. The bracketing groups C(J, J') and G(J, J') are defined in Section 1 below. In essence, C(J, J') is the mixed commutator group [E(A), E(J, J')], where E(J, J') is the E(A)-normalized subgroup generated by all elements x,(j) with a long and Received by the editors April 1, 1997 and, in revised form, May 22, 1997. 1991 Mathematics Sutbject Classification. Primary 20H05; Secondary 20G35. Research partially supported by NSA grant MDA 904-94-H-2008 and NSF grant DMS-9622899. (?)1999 American Mathematical Society