ω-elongations and Crawley’s problem

ω-elongations of Z(p) by separable p-primary groups are studied. Assuming (V = L), direct sums of cyclic groups are characterized using ω-elongations. Also assuming (V = L) much information is obtained about ω-elongations of Z(p) by groups which are not direct sums of cyclic groups. Finally it is shown that it is consistent that there is an uncountable group B with a countable basic subgroup such that there is a unique ω-elongation of Z(p) of B.