In a finite group, if the group and an element have the same order then the group is cyclic

Let G be a finite group and let x \in G. Prove that if |x| = |G| then G = \langle x \rangle. Give an explicit example to show that if G is infinite this need not be true.


Recall that |x| is short for |\langle x \rangle|. So in fact we have |G| = |\langle x \rangle|. Since \langle x \rangle \subseteq G and G is finite, G = \langle x \rangle.

Now consider r \in D_\infty; r has infinite order, so |r| = |D_\infty|. But D_\infty \neq \langle r \rangle since s \notin \langle r \rangle.

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