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Basic property of inverses of group elements of finite order


Let $G$ be a group and let $x \in G$. Prove that if $|x| = n$ for some $n \in \mathbb{Z}^+$, then $x^{-1} = x^{n-1}$.


Solution: We have $x \cdot x^{n-1} = x^n = 1$, so by the uniqueness of inverses $x^{-1} = x^{n-1}$.

Linearity

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