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I am currently in the process of proving that for a finite abelian group $G$, there exists a $g \in G$ with ord$(G) =$ exp$(G)$. I have everything I need, except the following statement:

Let $G$ be a finite group and $p$ be a prime divisor of exp$(G)$, such that we can write exp$(G) = p^f \cdot m$ with $f, m \in \mathbb N$ and $p, m$ coprime.

Show that there exists an element $g \in G$ with $p^f \mid \text{ord}(g)$.

I tried to work with the formula ord$(g^n) = \frac{\text{ord}(g)}{\text{gcd}(\text{ord}(g), n)}$ and the lcm, but I did not get anywhere. All help is therefore appreciated!

Arturo Magidin
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Minerva
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  • Show that if $g^{p^a}$ has order relatively prime to $p$ for all $g\in G$, then $f\leq a$. This will show that there must be an element with $g^{p^{f-1}}$ of order divisible by $p$, which allows you to obtain an element of order exactly $p^f$. – Arturo Magidin Oct 21 '23 at 17:50
  • Alternatively, this post shows it, since the exponent is the lcm of all orders. – Arturo Magidin Oct 21 '23 at 17:52

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