Does this result about sylow subgroups hold for infinite groups? (Rotman excercise) Let $P$ be a Sylow subgroup of $G$. Let $N(P) \leq H $. Prove $H=N(H)$. Did Rotman omit that $G$ is finite?
This is exercise 4.11 in the fourth edition.

Does it hold for infinite groups also?

Edit: I wrote a letter to Rotman and he told me I could assume the group is finite.

Rotman is obviously talking of finite groups here, as otherwise the meaning of Sylow subgroup would be very twisted (one can do that by extending the naturals and stuff). 
Yet the confusion is  understandble since Rotman all the times reminds the reader that a group is finite.
