About subgroups rich in transvections

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.