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Feb 10, 2015 at 9:37 comment added YCor Ah I see, because $y\mapsto [x,y]$ is a homomorphism from the inverse image of the centralizer of $x$ to the center.
Feb 10, 2015 at 9:19 comment added Derek Holt But they have index at most $2$ in the inverse image of the centralizer of $x$.
Feb 10, 2015 at 8:25 comment added YCor What do you mean? If you have $x$ and a lift $x'$, then clearly the centralizer of $x'$ is contained in the inverse image of the centralizer of $x$. But the other inclusion is unclear, could you elaborate?
Feb 10, 2015 at 8:07 vote accept Qiaochu Yuan
Feb 10, 2015 at 7:51 history answered Dima Pasechnik CC BY-SA 3.0