Let $(G,*)$ be a finite group and $x$ an element of $G$.
Prove that if $x^2=e$ then the order of $x$ can be only $1$ or $2$?
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You can consider two points as follows:
Well, the order of $x$ is defined as the least integer $n$ such that $x^n=e$. Since $x^2=e$, the order must be less than or equal to $2$, that is, $1$ or $2$.
I'm not sure I understand the question, this is the way I'm reading it: If $x^2=e$, is the order of $x$: $1$ or $2$? If this is it, then it depends on what is $x$:
We know that a group has a single unit, and being of order 1 means $x^1=x=e$, and therefore $e$ is the only element of order $1$, if $x\neq e$, then the order must be two (order of the subgroup it generates):
$$\left<x\right>=\left\lbrace x,x^2=e \right\rbrace$$
If the question is different, reformulate it and I will edit my answer.