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

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- Sep 7th 2007, 04:38 PMtttcomraderFinite group with element of prime order
Let G be a finite group with more than one element. Show that G has an element of prime order.

- Sep 7th 2007, 10:49 PMred_dog
Let $\displaystyle a\in G, \ ord(a)=n$.

If $\displaystyle n$ is prime, we've done.

Else, let $\displaystyle p$ be a prime divisor of $\displaystyle n$.

Then $\displaystyle a^n=a^{pm}=(a^m)^p=e$.

Let $\displaystyle b=a^m\in G\Rightarrow b^p=e$.

Let $\displaystyle 0<q<p$. Then $\displaystyle b^q=(a^m)^q=a^{mq}$.

But $\displaystyle mq<mp=n\Rightarrow a^{mq}\neq e$.

So $\displaystyle ord(b)=p$. - Sep 8th 2007, 03:26 AMtopsquark
- Sep 8th 2007, 04:51 PMThePerfectHacker