Bunina, E. I.; Mikhalev, A. V.; Roizner, M. A. A criterion for the elementary equivalence of endomorphism rings and automorphism groups of Abelian \(p\)-groups. (English. Russian original) Zbl 1306.20038 Dokl. Math. 90, No. 1, 399-400 (2014); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 457, No. 1, 11-12 (2014). From the text: We generalize results of Bunina and Roizner on automorphism groups and of Bunina and Mikhalev on endomorphism rings and obtain a criterion for the elementary equivalence of automorphism groups and endomorphism rings of Abelian \(p\)-groups in terms of a second order equivalence of these groups themselves. Cited in 2 Documents MSC: 20F28 Automorphism groups of groups 03C60 Model-theoretic algebra 03C85 Second- and higher-order model theory 20K10 Torsion groups, primary groups and generalized primary groups 20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups 20A15 Applications of logic to group theory 16S50 Endomorphism rings; matrix rings Keywords:automorphism groups; endomorphism rings; elementary equivalences; Abelian \(p\)-groups Citations:Zbl 1219.20026; Zbl 1073.16001 PDFBibTeX XMLCite \textit{E. I. Bunina} et al., Dokl. Math. 90, No. 1, 399--400 (2014; Zbl 1306.20038); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 457, No. 1, 11--12 (2014) Full Text: DOI References: [1] Baer, R., No article title, Ann. Math., 44, 192-227 (1943) · Zbl 0061.05405 · doi:10.2307/1968763 [2] I. Kaplansky, Infinite Abelian Groups (Univ. of Michigan Press, Ann Arbor, 1954; 1969). · Zbl 0057.01901 [3] Leptin, H., No article title, Math. Z., 73, 235-253 (1960) · Zbl 0093.24802 · doi:10.1007/BF01159715 [4] Liebert, W., No article title, Contemp. Math., 87, 51-59 (1989) · doi:10.1090/conm/087/995264 [5] Bunina, E. I.; Mikhalev, A. V., No article title, Fundam. Prikl. Mat., 10, 135-224 (2004) · Zbl 1073.20045 [6] L. Fuchs, Infinite Abelian Groups (Academic, New York, 1970, 1973; Mir, Moscow, 1974, 1977), Vols. 1, 2. · Zbl 0209.05503 [7] Roizner, M. A., No article title, Fundam. Prikl. Mat., 17, 157-163 (2011) [8] Roizner, M. A., No article title, Fundam. Prikl. Mat., 18, 159-170 (2013) [9] Shelah, S., No article title, Ann. Sci. Univ. Clermont, 13, 1-29 (1976) [10] Bragin, V. A.; Bunina, E. I., No article title, Fundam. Prikl. Mat., 18, 35-44 (2013) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.