Macedonska, O.; Slanina, Piotr \(GB\)-problem in the class of locally graded groups. (English) Zbl 1160.20021 Commun. Algebra 36, No. 3, 842-850 (2008). The authors consider a question which is equivalent to the following one posed by Bergman (1981): Let \(G\) be a group and \(S\) a subsemigroup of \(G\) which generates \(G\) as a group. Must each identity satisfied in \(S\) be satisfied in \(G\)? The first counterexample was found in 2005 by S. V. Ivanov and A. M. Storozhev [Proc. Am. Math. Soc. 133, No. 7, 1873-1879 (2005; Zbl 1076.20015)]. The authors of the article under review obtain a positive answer in the case of locally residually finite groups and for locally graded groups containing no free noncyclic subsemigroups. Reviewer: Igor Subbotin (Los Angeles) Cited in 2 Documents MSC: 20E10 Quasivarieties and varieties of groups 20E25 Local properties of groups 20F05 Generators, relations, and presentations of groups 20M05 Free semigroups, generators and relations, word problems Keywords:generating subsemigroups in groups; semigroup identities; group identities; locally residually finite groups; locally graded groups Citations:Zbl 1076.20015 PDFBibTeX XMLCite \textit{O. Macedonska} and \textit{P. Slanina}, Commun. Algebra 36, No. 3, 842--850 (2008; Zbl 1160.20021) Full Text: DOI References: [1] Bergman G., Aequat. Math. 23 pp 55– (1981) · Zbl 0453.08003 · doi:10.1007/BF02188011 [2] Bergman G., Questions in Algebra (1986) [3] Burns R. G., J. Algebra 195 pp 510– (1997) · Zbl 0886.20022 · doi:10.1006/jabr.1997.7088 [4] Černikov S. N., Dokl. Akad. Nauk SSSR 194 pp 1280– (1970) [5] Clifford A. H., Math. Surveys. (1964) [6] Clifford A. H., Math. Surveys. (1967) [7] Erschler A., J. Algebra 272 pp 154– (2004) · Zbl 1049.20019 · doi:10.1016/j.jalgebra.2002.11.005 [8] Hall P., Proc. London Math. Soc. 9 pp 595– (1959) · Zbl 0091.02501 · doi:10.1112/plms/s3-9.4.595 [9] Ivanov S. V., Proc. Amer. Math. Soc. 133 pp 1873– (2005) · Zbl 1076.20015 · doi:10.1090/S0002-9939-05-07903-7 [10] Krempa J., Contemporary Mathematics 131 pp 125– (1992) · doi:10.1090/conm/131.3/1175878 [11] Longobardi P., Trans. Amer. Math. Soc. 347 pp 1419– (1995) · doi:10.1090/S0002-9947-1995-1277124-5 [12] Lyndon R. C., Combinatorial Group Theory (1977) · Zbl 0368.20023 · doi:10.1007/978-3-642-61896-3 [13] Macedońska , O. ( 2003 ).Groupland. Vol. 305 . Lecture Note Ser. London Math. Soc. , pp. 400 – 404 . · Zbl 1085.20510 [14] Ol’shanskii A. Yu., J. Austral. Math. Soc. Series A 60 pp 255– (1996) · doi:10.1017/S1446788700037642 [15] Robinson D. J. S., A Course in the Theory of Groups (1982) · Zbl 0483.20001 · doi:10.1007/978-1-4684-0128-8 [16] Semple J. F., J. Algebra 157 pp 43– (1993) · Zbl 0814.20022 · doi:10.1006/jabr.1993.1089 [17] Shevrin L. N., Izv. Vyssh. Uchebn. Zaved. Mat. 6 pp 3– (1989) 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.