Geodesic
Extension of isomorphisms in abelian $p$-groups
Matematičeskie zametki, Tome 14 (1973) no. 4, pp. 543-548 
P. A. Krylov. Extension of isomorphisms in abelian $p$-groups. Matematičeskie zametki, Tome 14 (1973) no. 4, pp. 543-548. http://geodesic.mathdoc.fr/item/MZM_1973_14_4_a11/
@article{MZM_1973_14_4_a11,
     author = {P. A. Krylov},
     title = {Extension of isomorphisms in abelian $p$-groups},
     journal = {Matemati\v{c}eskie zametki},
     pages = {543--548},
     year = {1973},
     volume = {14},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1973_14_4_a11/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

It is proved that every isomorphism between any two subgroups of a group $G$ retaining the height of the elements in $G$ is extended to an automorphism of the group itself in the class of abelian $p$-groups without elements of infinite height if and only if $G$ is a closed group with finite Ulam invariants.