On overgroups of a cycle rich in transvections

A subgroup H of the general linear group G=GL(n,R) of order n over the ring R is said to be rich in transvections if it contains elementary transvections tij(α)=e+αeij at all positions (i,j), i≠j, for some α∈R, α≠0. This concept was introduced by Z. I. Borevich, considering the problem of describing subgroups of linear groups containing fixed subgroup. It is known that the overgroup of a nonsplit maximal torus containing an elementary transvection at some one position, is rich in transvections. For a commutative domain R with unit and a cycle π=(1 2 … n)∈Sn of length n, the following proposition is proved. A subgroup ⟨tij(α),(π)⟩ of the general linear group GL(n,R) generated by the permutation matrix (π) and the transvection tij(α) is rich in transvections if and only if the numbers i-j and n are coprime. A system of additive subgroups σ=(σij), 1≤i,j≤n, of a ring R is called a net (carpet) over a ring R of order n, if σirσrj⊆σij for all values of the indices i, r, j (Z. I. Borevich, V. M. Levchuk). The same system, but without the diagonal, called elementary net. We call a complete or elementary net σ=(σij) irreducible if all additive subgroups of σij are nonzero. In this note we define weakly saturated nets that play an important role in the proof of the main result.