Infinite group $G$ with subgroup $H$ such that $xHx^{-1} \subset H$, for some $x \in G$

Question

Give an example of $G$ an infinite group ($\#G=\infty$) having a subgroup $H$ such that $xHx^{-1} \subset H$, for some $x \in G$. (That is, $xHx^{-1} \neq H$.) [Hint: Take $G = S_z$, $H = \{f \in G : f(x) = x \text{ if } x < 0\}$.]

What exactly is $S_z$ here? The set of permutations of the integers? If so, I do not see how an example would follow from the hint. Could anyone point me in the right direction without spoiling the answer too much?

Answer 1

Without knowing more I couldn't be sure, but it seems fair to guess that $S_z$ is the permutation group of the integers.

What about the function $g(x)=x+1$ ? (I claim $gHg^{-1} \subsetneq H$.)

Explanation:

Then $gfg^{-1}(x)=gf(x-1)=g(x-1)=x$ for all nonpositive $x$, but we can't permute zero, so it's not all of $H$.