Exercise 1.1.5 ($\mathbb{Z}/n \mathbb{Z}$ is not a group under multiplication )

Question

Prove for all $n >1$ that $\mathbb{Z}/n \mathbb{Z}$ is not a group under multiplication of residues classes.

Richard Ganaye · Accepted Answer

\begin{proof} 
Let $n>1$ be an integer. Then $\overline{0} 
e \overline{1}$ (otherwise $0 \equiv 1 \pmod n$, so $n \mid 1$, therefore $|n| \leq 1$, in contradiction with $n>1$).
But $\overline{0} \cdot \overline{1} = \overline{0} = \overline{0} \cdot \overline{0}$. If $\mathbb{Z}/n \mathbb{Z}$ were a group, then  by Proposition 2 the equality $\overline{0} \cdot \overline{1} = \overline{0} \cdot \overline{0}$ implies $\overline{0} = \overline{1}$. This contradiction shows that $\mathbb{Z}/n \mathbb{Z}$ is not a group under multiplication.
\end{proof}

Exercise 1.1.5 ($\mathbb{Z}/n \mathbb{Z}$ is not a group under multiplication )

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Exercise 1.1.5 ($\mathbb{Z}/n \mathbb{Z}$ is not a group under multiplication )

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