A note on almost subnormal subgroups of linear groups

Abstract:

Following Hartley we say that a subgroup $H$ of a group $G$ is almost subnormal in $G$ if there is a series of subgroups $H = {H_0} \leqslant {H_1} \leqslant \cdots \leqslant {H_r} = G$ of $G$ of finite length such that for each $i < r$ either ${H_i}$ is normal in ${H_{i + 1}}$ or ${H_i}$ has finite index in ${H_{i + 1}}$. We extend a result of Hartley’s on arithmetic groups (see Theorem $2$ of Hartley’s Free groups in normal subgroups of unit groups and arithmetic groups, Contemp. Math., vol. 93, Amer. Math. Soc., Providence, RI, 1989, pp. 173-177) to arbitrary linear groups. Specifically, we prove: let $G$ be any linear group with connected component of the identity ${G^0}$ and unipotent radical $U$. If $H$ is any soluble-by-finite, almost subnormal subgroup of $G$ then $[H \cap {G^0},{G^0}] \leqslant U$.