Algebra Solutions: A group element and its inverse have the same order

Let $G$ be a group and let $x \in G$ . Prove that $x$ and $x^{-1}$ have the same order.

Recall that the order of a group element is either a positive integer or infinity.
Suppose $|x|$ is infinite and that $|x^{-1}| = n$ for some $n$ . Then $x^n = x^{-1 \cdot n \cdot -1} = ((x^{-1})^n)^{-1} = 1^{-1} = 1$ , a contradiction. So if $|x|$ is infinite, $|x^{-1}|$ must also be infinite. Likewise, if $|x^{-1}|$ is infinte, then $|(x^{-1})^{-1}| = |x|$ is also infinite.
Suppose now that $|x| = n$ and $|x^{-1}| = m$ are both finite. Then we have $(x^{-1})^n = (x^n)^{-1} = 1^{-1} = 1$ , so that $m \leq n$ . Likewise, $n \leq m$ . Hence $m = n$ and $x$ and $x^{-1}$ have the same order.

Algebra Solutions

A group element and its inverse have the same order

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