Endomorphism/Finite order/Minimal polynomial/Exercise

Let ${}\varphi \colon V\rightarrow V$ be a linear mapping on a finite-dimensional ${}K$-vector space ${}V$. Show that ${}\varphi$ has finite order if and only if the minimal polynomial of ${}\varphi$ is a divisor of ${}X^{n}-1$ for some ${}n\in \mathbb {N} _{+}$.

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