Geodesic
Elementary equivalence of the automorphism groups of Abelian $p$-groups
Fundamentalʹnaâ i prikladnaâ matematika, Tome 15 (2009) no. 7, pp. 81-112.

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

We consider Abelian $p$-groups ($p\geq3$) $A_1=D_1\oplus G_1$ and $A_2=D_2\oplus G_2$, where $D_1$ and $D_2$ are divisible and $G_1$ and $G_2$ are reduced subgroups. We prove that if the automorphism groups $\operatorname{Aut}A_1$ and $\operatorname{Aut}A_2$ are elementarily equivalent, then the groups $D_1$, $D_2$ and $G_1$, $G_2$ are equivalent, respectively, in the second-order logic. 
@article{FPM_2009_15_7_a2,
     author = {E. I. Bunina and M. A. Roizner},
     title = {Elementary equivalence of the automorphism groups of {Abelian} $p$-groups},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {81--112},
     publisher = {mathdoc},
     volume = {15},
     number = {7},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2009_15_7_a2/}
}
TY  - JOUR
AU  - E. I. Bunina
AU  - M. A. Roizner
TI  - Elementary equivalence of the automorphism groups of Abelian $p$-groups
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2009
SP  - 81
EP  - 112
VL  - 15
IS  - 7
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FPM_2009_15_7_a2/
LA  - ru
ID  - FPM_2009_15_7_a2
ER  - 
%0 Journal Article
%A E. I. Bunina
%A M. A. Roizner
%T Elementary equivalence of the automorphism groups of Abelian $p$-groups
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2009
%P 81-112
%V 15
%N 7
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FPM_2009_15_7_a2/
%G ru
%F FPM_2009_15_7_a2
E. I. Bunina; M. A. Roizner. Elementary equivalence of the automorphism groups of Abelian $p$-groups. Fundamentalʹnaâ i prikladnaâ matematika, Tome 15 (2009) no. 7, pp. 81-112. http://geodesic.mathdoc.fr/item/FPM_2009_15_7_a2/

[1] Bunina E. I., “Elementarnaya ekvivalentnost unitarnykh lineinykh grupp nad koltsami i telami”, Uspekhi mat. nauk, 53:2 (1998), 137–138 | DOI | MR | Zbl

[2] Bunina E. I., “Elementarnaya ekvivalentnost unitarnykh lineinykh grupp nad polyami”, Fundament. i prikl. mat., 4:4 (1998), 1265–1278 | MR | Zbl

[3] Bunina E. I., “Elementarnye svoistva grupp Shevalle”, Uspekhi mat. nauk, 56:1 (2001), 157–158 | DOI | MR | Zbl

[4] Bunina E. I., Mikhalëv A. V., “Elementarnaya ekvivalentnost kolets endomorfizmov abelevykh $p$-grupp”, Fundament. i prikl. mat., 10:2 (2004), 135–224 | MR | Zbl

[5] Bunina E. I., Mikhalëv A. V., “Elementarnye svoistva kategorii modulei nad koltsom, kolets endomorfizmov i grupp avtomorfizmov modulei”, Fundament. i prikl. mat., 10:2 (2004), 51–134 | MR | Zbl

[6] Keisler G., Chen Ch. Ch., Teoriya modelei, Mir, M., 1977 | MR

[7] Kulikov L. Ya., “K teorii abelevykh grupp proizvolnoi moschnosti”, Mat. sb., 9(51):1 (1941), 165–181 | MR | Zbl

[8] Kulikov L. Ya., “K teorii abelevykh grupp proizvolnoi moschnosti”, Mat. sb., 16(58):2 (1945), 129–162 | MR | Zbl

[9] Kulikov L. Ya., “Obobschënnye primarnye gruppy. I”, Tr. MMO, 1, 1952, 247–326 ; “II”, Тр. ММО, 2, 1953, 85–167 | MR | Zbl | MR

[10] Maltsev A. I., “Ob elementarnykh svoistvakh lineinykh grupp”, Problemy matematiki i mekhaniki, Novosibirsk, 1961, 110–132 | Zbl

[11] Fuks L., Beskonechnye abelevy gruppy, Mir, M., 1974

[12] Baer R., “Der Kern, eine charakteristische Untergruppe”, Compositio Math., 1 (1934), 254–283 | MR | Zbl

[13] Beidar C. I., Mikhalev A. V., “On Malcev's theorem on elementary equivalence of linear groups”, Contemp. Math., 131 (1992), 29–35 | DOI | MR | Zbl

[14] Boyer D. L., “On the theory of $p$-basic subgroups of Abelian groups”, Topics in Abelian Groups, Chicago, 1963, 323–330 | MR

[15] Erdélyi M., “Direct summands of Abelian torsion groups”, Acta Univ. Debrecen., 2 (1955), 145–149 | MR | Zbl

[16] Fuchs L., “On the structure of Abelian $p$-groups”, Acta Math. Acad. Sci. Hungar., 4 (1953), 267–288 | DOI | MR | Zbl

[17] Fuchs L., “Notes on Abelian groups. I”, Ann. Univ. Sci. Budapest, 2 (1959), 5–23 ; “II”, Acta Math. Acad. Sci. Hungar., 11 (1960), 117–125 | MR | Zbl | DOI | MR | Zbl

[18] Prüfer H., “Untersuchungen über die Zerlegbarkeit der abzählbaren primären Abelschen Gruppen”, Math. Z., 17 (1923), 35–61 | DOI | MR | Zbl

[19] Shelah S., “Interpreting set theory in the endomorphism semi-group of a free algebra or in the category”, Ann. Sci. Univ. Clermont Math., 13 (1976), 1–29 | MR | Zbl

[20] Szele T., “On direct decomposition of Abelian groups”, J. London Math. Soc., 28 (1953), 247–250 | DOI | MR | Zbl

[21] Szele T., “On the basic subgroups of Abelian $p$-groups”, Acta Math. Acad. Sci. Hungar., 5 (1954), 129–141 | DOI | MR | Zbl

[22] Tolstykh V., “Elementary equivalence of infinite-dimensional classical groups”, Ann. Pure Appl. Logic, 105 (2000), 103–156 | DOI | MR | Zbl