How hopficity is preserved under finite index? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T10:14:55Z http://mathoverflow.net/feeds/question/57444 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/57444/how-hopficity-is-preserved-under-finite-index How hopficity is preserved under finite index? Victor 2011-03-05T10:22:23Z 2011-03-05T19:14:40Z <p>Dear All! I guess this should be well-known, but I cannot find it in literature:</p> <p>Is it true that if \$H\$ is a subgroup of finite index in \$G\$, and \$H\$ is hopfian, then \$G\$ is also hopfian?</p> <p>In case when the groups are finitely generated, the answer is "yes" by R. Hirshon.</p> <p>(And the other way around question has answer no, as was kindly answered by Kate Juschenko, even for finitely generated case -- which is quite bizarre!!!)</p> http://mathoverflow.net/questions/57444/how-hopficity-is-preserved-under-finite-index/57461#57461 Answer by Kate Juschenko for How hopficity is preserved under finite index? Kate Juschenko 2011-03-05T14:10:52Z 2011-03-05T14:10:52Z <p>Baumslag-Solitar group BS(12,18) is Hopfian, but it contains normal subgroup of finite index which is not Hopfian. </p> <p>it is proved in the following paper:</p> <p>Baumslag; Solitar: Some two-generator one-relator non-Hopfian groups.</p>