On Shunkov groups with a strongly embedded almost layer-finite subgroup

Infinite Shunkov groups with the following condition are studied: the normalizer of any finite nontrivial subgroup has an almost layer-finite periodic part. Under this condition, the almost layer-finiteness of the periodic part of a Shunkov group with a strongly embedded almost layer-finite subgroup is established. Earlier, the author proved the almost layer-finiteness of a Shunkov group with a strongly embedded subgroup either under the condition that all proper subgroups are almost layer-finite or under the condition that the group is periodic. The case of a strongly embedded subgroup with a Chernikov almost layer-finite periodic part was also investigated earlier.