THE GROUPS WHOSE SUBGROUPS ARE ALMOST ASCENDANT

A subgroup H of a group G is called almost ascendant if H is ascendant in a subgroup K having finite index in G. We describe the structure of periodic groups whose subgroups are almost ascendant. The main result of this paper is following theorem. Let G be a periodic group whose subgroups are almost ascendant. Then G contains a normal finite subgroup K such that every subgroup of G/K is ascendant in G/K. In particular, G/K satisfies the normalizer condition.