The group of fractions of a semigroup S, if exists, can be written as G = SS−1. If S is abelian, then G must be abelian. We say that a semigroup identity is transferable if being satisfied in S it must be satisfied in G = SS−1. One of problems posed by G.Bergman in 1981 asks whether the group G must satisfy every semigroup identity which is satisfied in S, that is whether every semigroup identity is transferable. The first non-transferable identities were constructed in 2005 by S.V.Ivanov and A.M. Storozhev. A group G is called Hopfian if each epimorphizm G → G is the automorphism. The residually finite groups are Hopfian, however there are many problems concerning the Hopfian property e.g. of infinite Burnside groups, of finitely generated relatively free groups [11, Problem 15]. We prove here that if G = SS−1 is an n-generator group of fractions of a relatively free semigroup S, satisfying m-variable (m < n) non-transferable identity, then G is the non-Hopfian group.
@article{bwmeta1.element.doi-10_1515_math-2017-0037,
author = {Olga Macedo\'nska},
title = {On non-Hopfian groups of fractions},
journal = {Open Mathematics},
volume = {15},
year = {2017},
pages = {398-403},
zbl = {06715913},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_math-2017-0037}
}
Olga Macedońska. On non-Hopfian groups of fractions. Open Mathematics, Tome 15 (2017) pp. 398-403. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_math-2017-0037/