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 $t_{ij}(\alpha)=e+\alpha e_{ij}$ at all positions $(i, j)$, $i\neq j$, for some $\alpha\in R$, $\alpha\neq 0$. 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 $\pi=(1 \ 2 \ \ldots\ n)\in S_n$ of length $n$, the following proposition is proved. A subgroup $\langle t_{ij}(\alpha), (\pi) \rangle$ of the general linear group $GL(n, R)$ generated by the permutation matrix $(\pi)$ and the transvection $t_{ij}(\alpha)$ is rich in transvections if and only if the numbers $i-j$ and $n$ are coprime. A system of additive subgroups $\sigma=(\sigma_{ij})$, $1\leq i,j\leq n$, of a ring $R$ is called a net (carpet) over a ring $R$ of order $n$, if $\sigma_{ir} \sigma_{rj} \subseteq{\sigma_{ij}} $ 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 $\sigma = (\sigma_{ij})$ irreducible if all additive subgroups of $\sigma_{ij}$ are nonzero. In this note we define weakly saturated nets that play an important role in the proof of the main result.