Geodesic
Primary Abelian groups with isomorphic endomorphism groups
Matematičeskie zametki, Tome 14 (1973) no. 5, pp. 733-740

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A complete answer is given to the question, does isomorphism of the endomorphism groups of primary Abelian groups imply isomorphism of the groups themselves ? It is presumed that the generalized continuum hypothesis is valid. 
@article{MZM_1973_14_5_a13,
     author = {S. Ya. Grinshpon},
     title = {Primary {Abelian} groups with isomorphic endomorphism groups},
     journal = {Matemati\v{c}eskie zametki},
     pages = {733--740},
     publisher = {mathdoc},
     volume = {14},
     number = {5},
     year = {1973},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1973_14_5_a13/}
}
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S. Ya. Grinshpon. Primary Abelian groups with isomorphic endomorphism groups. Matematičeskie zametki, Tome 14 (1973) no. 5, pp. 733-740. http://geodesic.mathdoc.fr/item/MZM_1973_14_5_a13/