Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-03T02:48:14.375Z Has data issue: false hasContentIssue false

The normal and subnormal structure of general linear groups

Published online by Cambridge University Press:  24 October 2008

J. S. Wilson
Affiliation:
Christ's College, Cambridge

Extract

The problem of locating and classifying the normal subgroups of GLn(R), the general linear group of degree n over a commutative ring R with an identity element, has received considerable attention. The solution when R is a field is well known (of. Dieudonné(5), Artin(1)): unless n is equal to two and R has two or three elements, normal subgroups of GLn(R) either lie in the centre of GLn(R) or contain the special linear group SLn(R). However, if R is not a field, then for each ideal I of R the natural map RR/I induces a homomorphism

and, if 0 < I < R, the kernel of θ1 is a non-central normal subgroup of GLn(R) which does not contain SLn(R). The most that may be expected is that each normal subgroup determines an ideal I of R, in such a way that all normal subgroups determining the same ideal I lie between suitably defined greatest and smallest normal subgroups of GLn(R) corresponding to I. For example, write ZI for the inverse image of the centre of GLn(R/I) under the homomorphism θI, and write KI for the intersection of SLn(R) with the kernel of θI. Then KIZI, and the results of Klingenberg(7) and Mennicke(9) show that if n ≥ 3, and if R is either a local ring or the ring of rational integers, then any normal subgroup H of GLn(R) satisfies

for some uniquely determined ideal I. There are many similar theorems. That the above result breaks down for arbitrary rings, even for large n, follows easily from the negative solution to the congruence subgroup problem for certain rings of algebraic integers (see Bass, Milnor and Serre(3)).

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1972

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Artin, E.Geometric algebra (Interscience; New York–London, 1957).Google Scholar
(2)Bass, H.Algebraic K-theory (Benjamin; New York–Amsterdam, 1968).Google Scholar
(3)Bass, H., Milnor, J. and Serre, J.-P.Solution of the congruence subgroup problem for SLn(n ≽ 3) and Sp 2n (n 2). Inst. Hautes Études Sci. Publ. Math. 33 (1967), 59137.CrossRefGoogle Scholar
(4)Cohn, P. M.On the structure of the GL 2 of a ring. Inst. Hautes Études Sci. Publ. Math. 30 (1966), 553.CrossRefGoogle Scholar
(5)Dieudonné, J.La géométrie des groupes classiques (Springer-Verlag; Berlin–Gottingen–Heidelberg, 1955).Google Scholar
(6)Hall, P.A note on SI-groups. J. London Math. Soc. 39 (1964), 338344.CrossRefGoogle Scholar
(7)Klingenberg, W.Linear groups over local rings. Bull. Amer. Math. Soc. 66 (1960), 294296.CrossRefGoogle Scholar
(8)Kuroš, A. G. and Černikov, S. N.Soluble and nilpotent groups. Uspehi Mat. Nauk 2, No. 3(19) (1947), 1859. Amer. Math. Soc. Transl. No. 80 (1953).Google Scholar
(9)Mennicke, J. L.Finite factor groups of the unimodular group. Ann. of Math. (2) 81 (1965), 3137.CrossRefGoogle Scholar
(10)Merzijakov, Ju. I.On the theory of generalized solvable and nilpotent groups. (Russian.) Algebra i Logika Sem. 2 (1963), No. 5, 2936.Google Scholar
(11)Zassenhaus, H.The theory of groups (2nd ed., Chelsea; New York, 1958).Google Scholar