Is a normal subgroup of a finitely presented group finitely generated or normal finitely genrated? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T06:52:42Z http://mathoverflow.net/feeds/question/18974 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/18974/is-a-normal-subgroup-of-a-finitely-presented-group-finitely-generated-or-normal-f Is a normal subgroup of a finitely presented group finitely generated or normal finitely genrated? yeshengkui 2010-03-22T04:22:03Z 2010-03-22T04:35:11Z <p>Let \$G\$ be a finitely presented group and \$N\$ a normal subgroup. Is \$N\$ finitely generated or normally finitely generated? Here normally finitely generation means that for some finite set \$S\$ of elements in \$N\$, every element of \$N\$ can be writen as a product of \$G\$-conjugation of elements in \$S\$. Thanks. </p> http://mathoverflow.net/questions/18974/is-a-normal-subgroup-of-a-finitely-presented-group-finitely-generated-or-normal-f/18975#18975 Answer by Steve D for Is a normal subgroup of a finitely presented group finitely generated or normal finitely genrated? Steve D 2010-03-22T04:35:11Z 2010-03-22T04:35:11Z <p>If \$G\$ is the free group on two generators, then \$N\$ the commutator subgroup is not finitely generated. If \$H\$ is any finitely generated, but not finitely presented group, then \$H\$ is the quotient of a finitely generated free group \$G\$, with kernel \$N\$ which is not normally finitely generated.</p> <p>Steve</p>