Determining abelian $p$-groups from their $n$-socles. Mekler, A. H. & Shelah, S. Communications in Algebra, 18:287–307, 1990.
bibtex   
@article{MkSh:306,
author = {Mekler, Alan H. and Shelah, Saharon},
ams-subject = {(20K10)},
fromwhere = {3,IL},
journal = {Communications in Algebra},
review = {MR 91b:20074},
pages = {287--307},
title = {{Determining abelian $p$-groups from their $n$-socles}},
volume = {18},
year = {1990},
}
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