About Subgroups Rich in Transvections

Dzhusoeva, N. A. , Ikaev, S. S.  , Koibaev, V. A.
Vladikavkaz Mathematical Journal 2021. Vol. 23. Issue 4.

Abstract:
A subgroup \(H\) of the full 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 work is devoted to some questions associated with subgroups rich in transvections. It is known that if a subgroup \(H\) contains a permutation matrix corresponding to a cycle of length \(n\) and an elementary transvection of position \((i, j)\) such that \((i-j)\) and \(n\) are mutually simple, then the subgroup \(H\) is rich in transvections. In this note, it is proved that the condition of mutual simplicity of \((i-j)\) and \(n\) is essential. We show that for \(n=2k\), the cycle \(\pi=(1\ 2\ \ldots n)\) and the elementary transvection \(t_{31}(\alpha)\), \(\alpha\neq 0\), the group \(\langle (\pi), t_{31}(\alpha)\rangle\) generated by the elementary transvection \(t_{31}(\alpha)\) and the permutation matrix (cycle) \((\pi)\) is not a subgroup rich in transvections.

Keywords: subgroups rich in transvections, transvection, cycle

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