Algebra Solutions: ZZ/(n) is not a group under multiplication

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

Note that since $n > 1$ , $\overline{1} \neq \overline{0}$ . Now suppose $\mathbb{Z}/(n)$ contains a multiplicative identity element $\overline{e}$ . Then in particular, $\overline{e} \cdot \overline{1} = \overline{1}$ so that $\overline{e} = \overline{1}$ . Note, however, that $\overline{0} \cdot \overline{k} = \overline{0}$ for all $k$ , so that $\overline{0}$ does not have a multiplicative inverse. Hence $\mathbb{Z}/(n)$ is not a group under multiplication.

Algebra Solutions

ZZ/(n) is not a group under multiplication

No comments:

Post a Comment