On the normal subgroups of GL(2, A)

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In this paper we determine the normal subgroups of GL(2, A) for A a commutative local ring. The papers of Klingenberg [a], Lacroix [3], and our paper [ 1 ] constitute the previous work on this problem. As a consequence of our results, we are able to describe the normal subgroups of GL(2, A) for all Artinian rings A; hence, in particular, for all finite rings A. If N is a subset of GL(2, A), we denote by I(N) the level ideal of N, that is, the smallest ideal of A such that N consists of scalar matrices modulo I(N). For J an ideal of A, GC(2, A; J) = { TE GL(2, A) 1 I(T) E J> is the normal subgroup consisting of all matrices which are scalars modulo J, while SL(2, A; J) = { TE GL(2, A) 1 det T= 1 and Tz Imod J> is the principal congruence subgroup for J. Klingenberg proved that if i E A and the residue field of A is not G8’(3), then a subgroup N of GL(2, A) is normal if and only if SL(2, A; Z(N)) EN. Equivalently, N is normal if and only if SL(2, A; J) E Nc GC(2, A; J) for some ideal J, in which case J= Z(N). His paper also shows that this criterion for normality holds in GL(n, A) for n > 3 without any hypotheses on the local ring A. Our paper [ 1 ] characterizes normal subgroups of GL(2, A) for any %,-ring A with $6 A. Since every local ring is an SR,-ring, this leaves only the case of local rings in which 2 is not a unit, i.e., in which the residue field has characteristic 2. Lacroix, in [3], deals with local rings in which the residue field has characteristic 2 but is not GI;(2), obtaining the same normality criterion as Klingenberg. This leaves open the case of local rings with residue field GE(2). We shall use the “method of reduction” employed in Cl] to give a normality criterion for subgroups of GL(2, A) for A a local ring with GJ’(2) as residue field. Moreover, the method will allow us to recapture the results of Klingenberg and Lacroix with great efficiency. Thus, we will be able to 395 0021~8693/90 $3.00