Is there an infinite nilpotent group with one Sylow subgroup that is not normal?

Question

It is known that a finite nilpotent group has every Sylow subgroup normal in it. Does this result generalize to infinite nilpotent groups or not ?

If not, why, what is a counter example?

Answer 1

This is true, is not very complicated, and is part of Theorem 5.2.7 from Robinson's book "A Course in the Theory of Groups" (2 Edition):

5.2.7. Let $G$ be a nilpotent group. Then the elements of finite order in $G$ form a fully-invariant subgroup $Τ$ such that $G/T$ is torsion-free and $Τ = \operatorname{Dr}_p T_p$ where $T_p$ is the unique maximum $p$-subgroup of $G$.

$\operatorname{Dr}$ is the notation for the direct product in that book.