Algebra Solutions: Characterization of group elements whose square is the identity

Characterization of group elements whose square is the identity

Let $G$ be a group and let $x \in G$ . Prove that $x^2 = 1$ if and only if $|x|$ is either 1 or 2.

$(\Rightarrow)$ Suppose $x^2 = 1$ . Then we have $0 < |x| \leq 2$ , i.e., $|x|$ is either 1 or 2.
$(\Leftarrow)$ If $|x| = 1$ , then we have $x = 1$ so that $x^2 = 1$ . If $|x| = 2$ then $x^2 = 1$ by definition. So if $|x|$ is 1 or 2, we have $x^2 = 1$ .

Algebra Solutions

Characterization of group elements whose square is the identity

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