Geodesic
On definability of a periodic $\mathrm{EndE}^+$-group by its endomorphism group
Fundamentalʹnaâ i prikladnaâ matematika, Tome 13 (2007) no. 2, pp. 123-131 
E. M. Kolenova. On definability of a periodic $\mathrm{EndE}^+$-group by its endomorphism group. Fundamentalʹnaâ i prikladnaâ matematika, Tome 13 (2007) no. 2, pp. 123-131. http://geodesic.mathdoc.fr/item/FPM_2007_13_2_a3/
@article{FPM_2007_13_2_a3,
     author = {E. M. Kolenova},
     title = {On definability of a~periodic $\mathrm{EndE}^+$-group by its endomorphism group},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {123--131},
     year = {2007},
     volume = {13},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2007_13_2_a3/}
}
TY  - JOUR
AU  - E. M. Kolenova
TI  - On definability of a periodic $\mathrm{EndE}^+$-group by its endomorphism group
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2007
SP  - 123
EP  - 131
VL  - 13
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/FPM_2007_13_2_a3/
LA  - ru
ID  - FPM_2007_13_2_a3
ER  - 
%0 Journal Article
%A E. M. Kolenova
%T On definability of a periodic $\mathrm{EndE}^+$-group by its endomorphism group
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2007
%P 123-131
%V 13
%N 2
%U http://geodesic.mathdoc.fr/item/FPM_2007_13_2_a3/
%G ru
%F FPM_2007_13_2_a3

Voir la notice de l'article provenant de la source Math-Net.Ru

Let $\mathbf A$ be a class of Abelian groups, $A\in\mathbf A$, and $\mathrm{End}(A)$ be the additive endomorphism group of the group $A$. The group $A$ is said to be defined by its endomorphism group in the class $\mathbf B\supseteq\mathbf A$ if for every group $B\in \mathbf B$ such that $\mathrm{End}(B)\cong\mathrm{End}(A)$ the isomorphism $B\cong A$ holds. The paper considers the problem of definability of a periodic Abelian group $A$ such that $\mathrm{End}\bigl(\mathrm{End}(A)\bigr)\cong\mathrm{End}(A)$. The classes of periodical Abelian groups, of divisible Abelian groups, of reduced Abelian groups, of nonreduced Abelian groups, and of all Abelian groups are investigated in this paper.

[2] Grinshpon S. Ya., “Primarnye abelevy gruppy s izomorfnymi gruppami endomorfizmov”, Mat. zametki, 14:5 (1973), 733–741 | MR

[3] Grinshpon S. Ya., Sebeldin A. M., “Opredelyaemost periodicheskikh abelevykh grupp svoimi gruppami endomorfizmov”, Mat. zametki, 57:5 (1995), 663–669 | MR | Zbl

[4] Krylov P. A., Mikhalev A. V., Tuganbaev A. A., Svyazi abelevykh grupp i ikh kolets endomorfizmov, Tomsk, 2002

[5] Sebeldin A. M., “Vpolne razlozhimye abelevy gruppy bez krucheniya s izomorfnymi gruppami endomorfizmov”, Sbornik aspirantskikh rabot, Tomsk, 1976, 78–85

[6] Sebeldin A. M., “Ob opredelyaemosti abelevykh grupp bez krucheniya svoimi koltsami i gruppami endomorfizmov”, Gruppy i moduli, Tomsk, 1976, 78–85

[7] Sebeldin A. M., “Opredelyaemost neredutsirovannoi abelevoi gruppoi bez krucheniya svoei gruppoi endomorfizmov”, Abelevy gruppy i moduli, Tomsk, 1980, 102–108

[8] Fuks L., Beskonechnye abelevy gruppy, Mir, M., 1974; 1977

[9] Fuchs L., Abelian Groups, Budapest, 1958 | MR

[10] Sebeldin A. M., “Isomorphisme naturel des groupes des homomorphismes des groupes abéliens”, Ann. de L'IPGANC. Sér. A, 8 (1982), 155–158