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\(GB\)-problem in the class of locally graded groups. (English) Zbl 1160.20021

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.

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

Citations:

Zbl 1076.20015
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References:

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