Exercise 2.3.15 ($\mathbb{Q} imes \mathbb{Q}$ is not cyclic)

Question

Prove that $\mathbb{Q} 	imes \mathbb{Q}$ is not cyclic.

Richard Ganaye · Accepted Answer

\begin{proof}

Assume for the sake of contradiction that $\mathbb{Q} 	imes \mathbb{Q}$ is cyclic. Then every subgroup of $\mathbb{Q} 	imes \mathbb{Q}$ is cyclic. But $\mathbb{Z}	imes \mathbb{Z}$ is a subgroup of $\mathbb{Q} 	imes \mathbb{Q}$ which is not cyclic by Exercise 12 (c). Hence $\mathbb{Q} 	imes \mathbb{Q}$ is not cyclic.

\end{proof}
(Alternatively, we may copy and paste the proof of Exercise 12 (c), where we replace $\mathbb{Z} 	imes \mathbb{Z}$ by $\mathbb{Q} 	imes \mathbb{Q}$.)

Exercise 2.3.15 ($\mathbb{Q} \times \mathbb{Q}$ is not cyclic)

Answers

Comments

Exercise 2.3.15 ($\mathbb{Q} \times \mathbb{Q}$ is not cyclic)

Answers

Comments

Add answer