Complementability conditions for a periodic almost solvable subgroup in the group containing it

It is proved that if every prime Sylow subgroup of a periodic almost solvable (more generally, periodic W₀-) subgroup H of a group G has a complement in G and if, moreover, H is at most countable and the set (H) is finite, the subgroup H itself possesses a complement in G.Deceased.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 6, pp. 822–826, June, 1992.