Determining abelian p-Groups from their n-Socles

: A necessary condition is given for a very wide separable p-group to be determined by its n-socle (the set of elements of order at most pⁿ), A group G is very wide if it has a direct summand of final rank| G| which itself is a direct sum of cyclic groups. In the case, n = 1 (as Shelah has shown) this condition is sufficient but for n 5 1 examples are given to show that this condition is not sufficient.