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DOI: 10.46698/o2081-1390-1031-t

# 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