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Find an infinite group that has exactly two elements with order $4$?

Let $G$ be an infinite group for all $R_5$ (multiplication $\mod 5$) within an interval $[1,7)$. So $|2|=|3|=4$. Any other suggestions, please?

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    This is exercise 72 in chapter 4 of Gallian's Contemporary Abstract Algebra.2015-11-26

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