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DOI: 10.46698/o208113901031t 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 \((ij)\) 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 \((ij)\) 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
Language: Russian
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For citation: Dzhusoeva, N. A., Ikaev, S. S. and Koibaev, V. A. About Subgroups Rich in Transvections, Vladikavkaz Math. J., 2021, vol. 23, no. 4, pp.5055.
DOI 10.46698/o208113901031t ← Contents of issue 
 

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