Exercise 2.3.2 (If $G$ finite and $|x| = |G|$ then $G = \langle x \rangle$)

Question

If $x$ is an element of the finite group $G$ and $|x| = |G|$, prove that $G = \langle x \rangle$. Give an explicit example to show that this result need not be true if $G$ is an infinite group.

Richard Ganaye · Accepted Answer

\begin{proof} 
Suppose that $x$ is an element of the finite group $G$ and $|x| = |G|$. Then $|\langle x angle| = |G|$ and $\langle x angle \subseteq G$ where $G$ is finite, therefore $G = \langle x angle$.

As a counterexample if $G$ is infinite, take $G = \mathbb{Z}$ and $x = 2$. Then $|G| = |x| = \infty$, but $G = \mathbb{Z} 
e \langle 2 angle = 2 \mathbb{Z}$.
\end{proof}

Exercise 2.3.2 (If $G$ finite and $|x| = |G|$ then $G = \langle x \rangle$)

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Exercise 2.3.2 (If $G$ finite and $|x| = |G|$ then $G = \langle x \rangle$)

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