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May 28, 2012 at 10:02 comment added YCor you can say more: if $\langle a\rangle^G$ is torsion-free, it is equal to $\langle a\rangle$. Indeed, since it is normal and isomorphic to $\mathbf{Z}$, all its finite index subgroups remain normal (because all subgroups of $\mathbf{Z}$ are characteristic in $\mathbf{Z}$).
May 27, 2012 at 16:51 history answered Autumn Kent CC BY-SA 3.0