Timeline for About normal closure of cyclic subgroup

5 events

when toggle format	what		by	license	comment
May 28, 2012 at 10:58	vote	accept	Wei Zhou
May 28, 2012 at 10:07	comment	added	YCor		@Wei so what do you want? Kevin precisely answered your question. If you say you restrict to the torsion-free case, I understand your question as the following very interesting one "if $\langle a\rangle ^G$ is torsion-free, are all its nontrivial elements of infinite order" :) anyway indeed there is no nontrivial torsion-free example, see Richard's answer and my comment.
May 28, 2012 at 5:27	comment	added	Wei Zhou		Thank you very much. It answered my question. I don't take the answer as the best only because I want to know the case that $\langle a \rangle ^G$ is torsion-free.
May 28, 2012 at 2:54	comment	added	Misha		More generally, take a finite group F and a semidirect product $F\rtimes Z=G$ which is not split as a direct product. Then there will be an element g∈G which does not normalize Z=⟨a⟩. The groups $Z^g$ and $Z$ generate a subgroup of G containing nontrivial finite order elements.
May 27, 2012 at 18:30	history	answered	Kevin Ventullo	CC BY-SA 3.0