On Certain Applications of the Khukhro-Makarenko Theorem

Some recent results of Khukhro and Makarenko on the existence of characteristic X-subgroups of finite index in a group G, for certain varieties X, are used to obtain generalisations of some well-known results in the literature pertaining to groups G, in which all proper subgroups satisfy some condition or other related to the property 'soluble-by-finite'. In addition, a partial generalisation is obtained for the aforementioned results on the existence of characteristic subgroups.