##
**The \(E(2,A)\) sections of \(SL(2,A)\).**
*(English)*
Zbl 0743.20048

The classification of the normal subgroups of the classical linear groups over rings (e.g. general linear, symplectic) is a problem which has attracted much attention over many years. For any commutative ring \(R\) those subgroups of \(GL(n,R)\) which are normalized by\(E(n,R)\), the subgroup generated by the elementary matrices, can be completely classified in terms of the \(R\)-ideals, provided \(n\geq 3\). This classification was finally completed by Vaserstein after many special cases had previously been considered. (Brenner dealt with the first non- trivial case, namely \(R=\mathbb{Z}\), the ring of rational integers.)

For \(n=2\) the extremely complicated normal subgroup structure of \(SL(2,\mathbb{Z})\) (\(=E(2,\mathbb{Z})\)) indicates that it is unlikely that in general the \(E(2,R)\)-normalized subgroups of \(SL(2,R)\) can be satisfactorily classified. For any kind of reasonable classification to be possible it appears that the ring \(R\) has to contain “many” units.

In this paper the authors classify the \(E(2,A)\)-normalized subgroups of \(SL(2,A)\), where \(A\) is a ring of two types. Either \(A\) is an \(SR_ 2\)- ring or \(A\) is a subring, containing a unit of infinite order, of the algebraic closure of \(\mathbb{Q}\), the set of rational numbers, or \(k(x)\), the field of rational functions over a finite field \(k\). This extends many known results. From previous results it is known that subgroups can be completely classified in terms of the \(A\)-ideals, provided \(1/2\in A\). Considerable complications arise however when 2 is not a unit. Menal and Vaserstein have classified the \(E(2,L)\)-normalized subgroups of \(SL(2,L)\), where \(L\) is a local ring not containing \(1/2\), whose residue field has at least 4 elements. Their classification (which extends to non-commutative \(L\)) depends upon an algebraic structure (generalizing the notion of an ideal) which they call a quasi-ideal. (Every quasi-ideal is an ideal when 2 is a unit.) Prior to the paper under review the case when \(A\) has a maximal ideal of index 2 had not been considered.

In order to deal with this case the authors introduce a new algebraic structure which they call a radix. They classify the \(E(2,A)\)-normalized subgroups of \(SL(2,A)\) (in full generality) in terms of radices. Their results extend all the known previous results since (a) every radix is an ideal when \(1/2\in A\) and (b) every radix is a quasi-ideal when \(A\) has no maximal ideal of index 2.

For \(n=2\) the extremely complicated normal subgroup structure of \(SL(2,\mathbb{Z})\) (\(=E(2,\mathbb{Z})\)) indicates that it is unlikely that in general the \(E(2,R)\)-normalized subgroups of \(SL(2,R)\) can be satisfactorily classified. For any kind of reasonable classification to be possible it appears that the ring \(R\) has to contain “many” units.

In this paper the authors classify the \(E(2,A)\)-normalized subgroups of \(SL(2,A)\), where \(A\) is a ring of two types. Either \(A\) is an \(SR_ 2\)- ring or \(A\) is a subring, containing a unit of infinite order, of the algebraic closure of \(\mathbb{Q}\), the set of rational numbers, or \(k(x)\), the field of rational functions over a finite field \(k\). This extends many known results. From previous results it is known that subgroups can be completely classified in terms of the \(A\)-ideals, provided \(1/2\in A\). Considerable complications arise however when 2 is not a unit. Menal and Vaserstein have classified the \(E(2,L)\)-normalized subgroups of \(SL(2,L)\), where \(L\) is a local ring not containing \(1/2\), whose residue field has at least 4 elements. Their classification (which extends to non-commutative \(L\)) depends upon an algebraic structure (generalizing the notion of an ideal) which they call a quasi-ideal. (Every quasi-ideal is an ideal when 2 is a unit.) Prior to the paper under review the case when \(A\) has a maximal ideal of index 2 had not been considered.

In order to deal with this case the authors introduce a new algebraic structure which they call a radix. They classify the \(E(2,A)\)-normalized subgroups of \(SL(2,A)\) (in full generality) in terms of radices. Their results extend all the known previous results since (a) every radix is an ideal when \(1/2\in A\) and (b) every radix is a quasi-ideal when \(A\) has no maximal ideal of index 2.

Reviewer: A.W.Mason (Glasgow)

### MSC:

20H25 | Other matrix groups over rings |

20E07 | Subgroup theorems; subgroup growth |

20E15 | Chains and lattices of subgroups, subnormal subgroups |

20H05 | Unimodular groups, congruence subgroups (group-theoretic aspects) |

20G35 | Linear algebraic groups over adèles and other rings and schemes |