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- #1

I suspect not. However, I cannot seem to find a counter-example.

(By "a transversal for $G/H$" I mean that

1) $g^nH=g^mH\Rightarrow m=n$

2) if $hH$ is a coset of $G/H$ then there exists some $n\in\mathbb{Z}$ such that $g^nh^{-1}\in H$

so the powers of $g$ form a set of coset representatives for $G/H$, and no two of these representatives lie in the same coset.)