Being $o(x)\mid o(y)$, can we say that $x$ is a recurrence of $y$ (if $x$ is the power of $y$)?

Question

I've got a non cyclic group called G and x and y two elements from G. And $o(x)\mid o(y)$. Can we say that $x$ is a recurrence of $y$?

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I think that we can't but I don't really know how to justificate it.

Answer 1

HINT

$C_2\times C_2=\{1,x,y,xy|\ x^2=y^2=xyx^{-1}y^{-1}=1\}$ Then $o(x)|o(y)$ but $y^k\in \{1,y\}$