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Let $G$ be a finite group with more than one element. Show that $G$ has an element of prime order

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Does Z4 (the mod 4 group) have an element of prime order? The identity has order 1, and the other elements have order 4. – barrycarter Jan 25 '13 at 14:01
@barrycarter No, $2$ has order $2$. – SpamIAm Jun 27 '15 at 14:32
I can't believe I made such a simple error. – barrycarter Jul 4 '15 at 3:08

Hint: Let $a \neq 1$ be an element of $G$ of order $k$ and $d \in \mathbb{N}$ be a number such that $d \mid k$. What is the order of $a^d$?

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+1 You really show the OP how to find that element in $G$. ;-) – Babak S. Jan 25 '13 at 8:59

Let $|G|=n$ and $p\mid n$ such that $p$ be a prime , so it is enough to apply Cauchy's Theorem and find that desired element.

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Nice observation! +1 – amWhy Feb 2 '13 at 0:16
@amWhy: Thanks for your kindness. – Babak S. Feb 2 '13 at 6:25

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