If $x$ is the only element that $x^2=e$ then $x\in Z(G)$ $G$ be a group.
$$Z(G)=\{u\in G\mid ua=au \quad \forall a\in G \}$$
If $x$ is the only element in $G$ that satisfies  $x^2=e$ then $x\in Z(G)$
Attempt:

*

*$x^2=e$ then $(\forall g\in G),\; gb^2=g=b^2g$ then $gb=b^2gb^{-1}=gb=gb^{-1}\ldots$
it is not good.


*I considered cayley table to examine elements but it did not go well.


*I considered conjugate things but couldnot create a reasonable way to solve this.

Hint: $o(gxg^{-1})=o(x)$.
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