Locally soluble-by-finite groups with small deviation for non-subnormal subgroups

A group G has subnormal deviation at most 1 if, for every descending chain H0 > H1 > . . . of non-subnormal subgroups of G, for all but finitely many i there is no infinite descending chain of non-subnormal subgroups of G that contain Hi+1 and are contained in Hi. This property P, say, was investigated in a previous paper by the authors, where soluble groups with P and locally nilpotent groups with Pwere effectively classified. The present article affirms a conjecture from that article by showing that locally soluble-by-finite groups with P are soluble-by-finite and are therefore classified.