Almost layer finiteness of a periodic group without involutions

We prove a theorem that characterizes the class of almost layer finite groups in the class of periodic groups without involutions: If the normalizer of any nontrivial finite subgroup of a periodic conjugate biprimitive finite group without involutions is almost layer finite, then the group itself is almost layer finite. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 11, pp. 1529–1533, November, 1999.