On Overgroups of a Cycle Rich in Transvections

Dryaeva, R. Y.
Vladikavkaz Mathematical Journal 2024. Vol. 26. Issue 1.

Abstract:
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.

Keywords: subgroups rich in transvections, transvection, cycle, net, net group.

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