Exercise 1.1.16 ($x^2 = 1$ if and only if $|x|$ is either $1$ or $2$)

Question

Let $x$ be an element of $G$. Prove that $x^2 = 1$ if and only if $|x|$ is either $1$ or $2$.

Richard Ganaye · Accepted Answer

\begin{proof} 
If $x^2 = 1$, then the least positive integer such that $x^2 = 1$ is $1$ or $2$ so by definition $|x| 1$ or $|x| = 2$.

Conversely, if $x^2 = 1$, then $|x|= 2$ if $x 
e 1$, and $|x| = 1$ if $x = 1$.

For all $x \in G$, $x^2 = 1$ if and only if $|x|$ is either $1$ or $2$.
\end{proof}

Exercise 1.1.16 ($x^2 = 1$ if and only if $|x|$ is either $1$ or $2$)

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Exercise 1.1.16 ($x^2 = 1$ if and only if $|x|$ is either $1$ or $2$)

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