A true relative of Suslin's normality theorem (Q5963023)

Let \(A\) be a unital commutative ring and \(n \geq 2\). Then \(E_n(A)\) denotes the subgroup of \(\mathrm{SL}_n(A)\) that is generated by elementary matrices. Suslin's Normality Theorem states that \(E_n(A)\) is normal in \(\mathrm{SL}_n(A)\) (\(n \geq 3\)) [\textit{A. A. Suslin}, Math. USSR, Izv. 11, 221--238 (1977; Zbl 0378.13002)]. The article under review focuses on subgroups of \(\mathrm{SL}_n(A)\) generated by elementary matrices whose off diagonal non-zero entry belongs to an ideal \(\pi\) of \(A\). More precisely, there are two subgroups defined for an ideal \(\pi\) of \(A\). The first one is \(E_n({\pi})\) which is the normal subgroup of \(E_n(A)\) that is generated by elementary matrices with the non-zero entry belongs to \(\pi\). The second one is \(F_n({\pi})\) (the true relative elementary group) is as the first one but it is not required to be normal in \(E_n(A)\).NEWLINENEWLINEWe assume that \(n \geq 3\). Suslin generalized his normality theorem and proved that \(E_n({\pi})\) is normal in \(\mathrm{SL}_n(A)\). The main result of the paper is that \(F_n({\pi})\) is normal in the subgroup of \(\mathrm{SL}_n(A)\) consisting of those matrices which are diagonal mod\({\pi}\). Tits' result ([\textit{J. Tits}, C. R. Acad. Sci., Paris, Sér. A 283, 693--695 (1976; Zbl 0408.20028)]) asserts that \(E_n({\pi}^2)\) is a subgroup of \(F_n({\pi})\). The author proves Suslin's, Tits' and his main result directly using the decomposition of the conjugates of elementary matrices described by Suslin.NEWLINENEWLINEThe author presents the history of the important spacial case when \(A = \mathbb{Z}\). In this case, the results of Mennicke are central. The early result states that \(E_n(N) = {\Gamma}_n(N)\), the principal congruence subgroup ([\textit{J. L. Mennicke}, Ann. Math. (2) 81, 31--37 (1965; Zbl 0135.06504)]). He also proved that any normal subgroup of \(\mathrm{SL}_n(\mathbb{Z})\) is either central or it contains a principal congruence subgroup. Much later, Mennicke generalized his result and showed that \(F_n(N) = {\Delta}_n(N)\), the congruence subgroup of matrices in \({\Gamma}_n(N)\) whose diagonal entries are \(1\text{mod}\; N^2\). It was shown that the two statements are essentially equivalent.NEWLINENEWLINEIn the next section, the author discusses the relation between elementary and congruence subgroups. His main result is that the inclusions induce surjections of groups: NEWLINE\[NEWLINE{\Gamma}_n({\pi}^2)/E_n({\pi}^2) \twoheadrightarrow {\Delta}_n({\pi})/F_n({\pi}) \twoheadrightarrow {\Gamma}_n({\pi})/E_n({\pi})NEWLINE\]NEWLINE The main result of the paper shows that the middle set is a group. The Bass-Milnor-Serre theory states that, for \(A\) a ring of integers in a number field \(K\), establishes that \({\Gamma}_n({\pi})/E_n({\pi})\) is a finite cyclic group whose order divides the number of roots of unity in \(K\) and it is trivial if \(K\) admits a real embedding. The same is true for the middle group.NEWLINENEWLINEFor \(n = 2\), the author proves that \(F_2(N)\) is not normal in \({\Delta}_2(N)\) and \(E(N^2)\) is not contained in \(F_2(N)\)m, for \(N \geq 4\). This is contrary to Varenstein's result who proved the corresponding inclusions for an ideal \(\pi\) in \(A\) a ring of integers in a number field \(K\) which is not either \(\mathbb{Q}\) or \(\mathbb{Q}{[\sqrt{-D}]}\).