'Cojaments, suggestions, and related referenoes »eloomedJ Subgroups close to normal subgroups

Let G be a group and H a subgroup. It is shown that the set of indices {[H:HngHg~] | geG) has a finite upper bound n if and only if there is a normal subgroup N < G which is commensurable with //; i.e., such that [H:Nr\H] and [N:Nr>H] are finite; moreover, the latter indices admit bounds depending only on n. If the bounded index hypothesis is assumed only for g running over some subgroup K} for p prime. In particular, when p = 2 there exists N < G such that either H has index 2 in 7V, or N has index 2 in H.