A NOTE ON ALMOST SUBNORMAL SUBGROUPS OF LINEAR-GROUPS

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 = H0 less-than-or-equal-to H1 less-than-or-equal-to ... less-than-or-equal-to 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 G0 and unipotent radical U. If H is any soluble-by-finite, almost subnormal subgroup of G then [H and G0, G0] less-than-or-equal-to U.