Geodesic
Abelian groups isomorphic to a~proper fully invariant subgroup
Fundamentalʹnaâ i prikladnaâ matematika, Tome 22 (2019) no. 5, pp. 29-53.

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

The present paper is a survey of the authors' results related to studying groups containing a proper fully invariant subgroup isomorphic to the group. 
@article{FPM_2019_22_5_a4,
     author = {S. Ya. Grinshpon and M. M. Nikolskaya},
     title = {Abelian groups isomorphic to a~proper fully invariant subgroup},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {29--53},
     publisher = {mathdoc},
     volume = {22},
     number = {5},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2019_22_5_a4/}
}
TY  - JOUR
AU  - S. Ya. Grinshpon
AU  - M. M. Nikolskaya
TI  - Abelian groups isomorphic to a~proper fully invariant subgroup
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2019
SP  - 29
EP  - 53
VL  - 22
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FPM_2019_22_5_a4/
LA  - ru
ID  - FPM_2019_22_5_a4
ER  - 
%0 Journal Article
%A S. Ya. Grinshpon
%A M. M. Nikolskaya
%T Abelian groups isomorphic to a~proper fully invariant subgroup
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2019
%P 29-53
%V 22
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FPM_2019_22_5_a4/
%G ru
%F FPM_2019_22_5_a4
S. Ya. Grinshpon; M. M. Nikolskaya. Abelian groups isomorphic to a~proper fully invariant subgroup. Fundamentalʹnaâ i prikladnaâ matematika, Tome 22 (2019) no. 5, pp. 29-53. http://geodesic.mathdoc.fr/item/FPM_2019_22_5_a4/

[1] Grinshpon S. Ya., “O nekotorykh klassakh primarnykh abelevykh grupp, pochti izomorfnykh po vpolne kharakteristicheskim podgruppam”, Izv. vyssh. uchebn. zaved. Matematika, 1976, no. 2, 23–30 | Zbl

[2] Grinshpon S. Ya., “O stroenii vpolne kharakteristicheskikh podgrupp abelevykh grupp bez krucheniya”, Abelevy gruppy i moduli, 1982, 56–92

[3] Grinshpon S. Ya., “Vpolne kharakteristicheskie podgruppy abelevykh grupp i vpolne tranzitivnost”, Fundament. i prikl. matem., 8:2 (2002), 407–473 | MR | Zbl

[4] Grinshpon S. Ya., Nikolskaya M. M., “Abelevy $p$-gruppy, neizomorfnye sobstvennym vpolne kharakteristicheskim podgruppam”, Algebra, logika i prilozheniya, Tez. dokl., Krasnoyarsk, 2010, 27–28

[5] Grinshpon S. Ya., Nikolskaya M. M., “Sobstvennye vpolne kharakteristicheskie podgruppy, izomorfnye gruppe”, Abelevy gruppy, Materialy Vseros. simp., posvyaschennogo 95-letiyu L. Ya. Kulikova (Biisk, 19–25 avgusta 2010), Biisk, 2010, 29–31

[6] Grinshpon S. Ya., Nikolskaya M. M., “Sobstvennye vpolne kharakteristicheskie podgruppy grupp bez krucheniya, izomorfnye samoi gruppe”, Vestn. Tomskogo gos. un-ta. Ser. Matematika i mekhanika, 2012, no. 1(17), 25–30

[7] Grinshpon S. Ya., Nikolskaya (Savinkova) M. M., “$\mathrm{IF}$-gruppy”, Vestn. Tomskogo gos. un-ta. Ser. Matematika i mekhanika, 2010, no. 1(9), 5–14

[8] Grinshpon S. Ya., Nikolskaya M. M., “Primarnye $\mathrm{IF}$-gruppy”, Vestn. Tomskogo gos. un-ta. Ser. Matematika i mekhanika, 2011, no. 3(15), 25–31

[9] Kulikov L. Ya., “K teorii abelevykh grupp proizvolnoi moschnosti”, Matem. sb., 1941, no. 9, 165–182

[10] Kulikov L. Ya., “K teorii abelevykh grupp proizvolnoi moschnosti”, Matem. sb., 1945, no. 16, 129–162 | Zbl

[11] Nikolskaya M. M., “$\mathrm{IF}$-gruppy bez krucheniya”, Sovremennye problemy matematiki i mekhaniki, Materialy II Vseros. molodezhnoi nauchnoi konf., Izd-vo Tomskogo un-ta, Tomsk, 2011, 33–35

[12] Savinkova M. M., “Vpolne kharakteristicheskie podgruppy abelevykh $p$-grupp, izomorfnykh samoi gruppe”, Mezhdunar. algebraicheskaya konf., posvyaschennaya 100-letiyu so dnya rozhdeniya A. G. Kurosha, tez. dokl., M., 2008, 200–201

[13] Savinkova M. M., “O separabelnykh $p$-gruppakh, soderzhaschikh vpolne kharakteristicheskie podgruppy, izomorfnye samoi gruppe”, Vseros. konf. po matematike i mekhanike, tez. dokl. (22–25 sentyabrya 2008), Tomsk, 2008, 62

[14] Savinkova M. M., “$U$-posledovatelnosti i primarnye gruppy, soderzhaschie sobstvennye izomorfnye sebe vpolne kharakteristicheskie podgruppy”, Vestn. Tomskogo gos. un-ta. Ser. Matematika i mekhanika, 2008, no. 2(3), 56–60

[15] Fuks L., Beskonechnye abelevy gruppy, v. 1, Mir, M., 1974

[16] Fuks L., Beskonechnye abelevy gruppy, v. 2, Mir, M., 1977

[17] Sherstneva (Botygina) A. I., “$U$-posledovatelnosti i pochti izomorfizm abelevykh $p$-grupp po vpolne kharakteristicheskim podgruppam”, Izv. vyssh. uchebn. zaved. Matematika, 2001, no. 5, 72–80 | MR | Zbl

[18] Beaumont R. A., “Groups with isomorphic proper subgroups”, Bull. Amer. Math. Soc., 51 (1945), 381–387 | DOI | MR | Zbl

[19] Beaumont R. A., Pierce R. S., “Partly transitive modules and modules with proper isomorphic submodules”, Trans. Amer. Math. Soc., 91 (1959), 209–219 | DOI | MR | Zbl

[20] Beaumont R. A., Pierce R. S., “Isomorphic direct summands of Abelian groups”, Math. Ann., 153 (1964), 21–37 | DOI | MR

[21] Benabdallah K. M., Eisenstadt B. J., Irwin J. M., Poluianov E. W., “The structure of large subgroups of primary Abelian groups”, Acta Math. Acad. Sci. Hung., 21:3–4 (1970), 421–435 | DOI | MR | Zbl

[22] Crawley P., “An infinite primary Abelian group without proper isomorphic subgroups”, Bull. Amer. Math. Soc., 68 (1962), 463–467 | DOI | MR | Zbl

[23] Goldsmith B., Óhógáin S., Wallutis S., “Quasi-minimal groups”, Proc. Amer. Math. Soc., 132:8 (2004), 2185–2195 | DOI | MR | Zbl

[24] Grinshpon S. Ya., Krylov P. A., “Fully invariant subgroups, full transitivity and homomorphism groups of Abelian groups”, J. Math. Sci., 128:3 (2005), 2894–2997 | DOI | MR | Zbl

[25] Grinshpon S. Ya., Nikolskaya (Savinkova) M. M., “Fully invariant subgroups of Abelian $p$-groups with finite Ulm–Kaplansky invariants”, Commun. Algebra, 39:11 (2011), 4273–4282 | DOI | MR | Zbl

[26] Hill P., Megibben Ch., “On primary groups with countable basic subgroups”, Trans. Amer. Math. Soc., 124:1 (1966), 49–59 | DOI | MR | Zbl

[27] Kaplansky I., Infinite Abelian groups, Univ. Michigan Press, Ann Arbor, 1968 | MR

[28] Monk G. S., “Abelian $p$-groups without proper isomorphic pure dense subgroups”, Illinois J. Math., 14:1 (1970), 164–177 | DOI | MR | Zbl

[29] Pierce R., “Homomorphisms of primary Abelian groups”, Topics in Abelian Groups, Chicago, 1963, 215–310 | MR