An elementary account of Selberg’s lemma. (English) Zbl 0639.20030

Alperin, Roger C.

Enseign. Math., II. Sér. 33, 269-273 (1987).

The author offers proofs of the following known results. Let G be a subgroup of GL(n,R), where R is a finitely generated integral domain of characteristic \(p\geq 0\). a) G is residually finite. b) If \(p=0\) then G is torsion-free by finite. c) If \(p>0\) then G is a finite extension of a group whose finite elements are all unipotent.
a) is due to A. I. Mal’cev [Mat. Sb., Nov. Ser. 8, 405-422 (1940; Zbl 0025.00804) = Transl., II. Ser., Am. Math. Soc. 45, 1-18 (1965)], b) appears in A. Selberg [Contrib. Function Theory, Tata Inst. Bombay 1960, 147-164 (1960; Zbl 0201.366)], effectively in Yu. I. Merzlyakov [Algebra Logika 3, No. 4, 49-59 (1964; Zbl 0244.20048)] and in M. I. Kargapolov [Algebra Logika 6, No. 5, 17-20 (1967; Zbl 0252.20042)] and the obvious variant c) of b) in V. P. Platonov [Dokl. Akad. Nauk BSSR 12, 492-494 (1968; Zbl 0228.20018)] and the reviewer [Proc. Lond. Math. Soc., III. Ser. 20, 101-122 (1970; Zbl 0188.062)].
a) is an easy consequence of Hilbert’s Nullstellensatz (which can be dressed up as field theory of course) and b) and c) of the Krull intersection theorem. Here the author presents ‘field theoretic’ proofs of these results. The basic trick is to reduce to R being a suitable localization of a polynomial ring, a trick that essentially also appears in Mal’cev’s original 1940 paper. This reduction looses information about the group G and hence usually needs to be avoided.

Reviewer: B.A.F.Wehrfritz

MSC:

20H20	Other matrix groups over fields
20E26	Residual properties and generalizations; residually finite groups
20G35	Linear algebraic groups over adèles and other rings and schemes

Keywords:

subgroup of GL(n,R); finitely generated integral domain; residually finite; torsion-free by finite

Citations:

Zbl 0025.00804; Zbl 0201.366; Zbl 0244.20048; Zbl 0252.20042; Zbl 0228.20018; Zbl 0188.062