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### Problem 1

**(a)** Show that multiplication is a well-defined binary operation on the set $\mathbb{Z_n}$ of congruence classes of integers modulo $n$.

**(b)** Given an integer $n \ge 1$, let $\mathbb{Z^{*}_{n}}$ be the set of elements $\bar x \in \mathbb{Z_{n}}$ such that there exists $\bar y \in \mathbb{Z_{n}}$ with $\bar x \bar y = \bar 1$. Show that $\mathbb{Z^{*}_{n}}$ with the operation of multiplication is a group.

**(c)** Write multiplication tables for $\mathbb Z^*_8$,$\mathbb Z^*_{10}$ and $\mathbb Z^*_{12}$.

**(d)** Show that $\mathbb{Z^{*}_{8}} \simeq \mathbb{Z^{*}_{12}}$, but that $\mathbb{Z^{*}_{10}}$ is not isomorphic to $\mathbb{Z^{*}_{8}}$ and $\mathbb{Z^{*}_{12}}$.

**Solutions**

**(a)**

For each $(a,b)\in \mathbb{Z_n}\times \mathbb{Z_n}$, when $a\times b\ge n$, $a*b=a\times b-n\in\mathbb{Z_n}$. When $a\times b<n$,$a*b=a\times b\in\mathbb{Z_n}$. So multiplication is a well-defined binary operation on the set $\mathbb{Z_n}$.

This just gives the definition of the operation; it doesn't show it's well-defined.

**(b)**

$\mathscr{G_{1}}$: Let $\bar x, \ \bar y, \ \bar z \in \mathbb{Z_{n}}$.

Now,

(1)where the middle equality holds since multiplication is associative in $\mathbb{Z}$.

Hence, multiplication is associative in $\mathbb{Z}_n^*$.

$\mathscr{G_{2}}$: Note $1 \cdot 1 = 1$, so $1 \in \mathbb{Z}_n^*$. Let $x \in \mathbb{Z_{n}}$. Now, $x \cdot 1 = 1 \cdot x = x$. Thus, $1$ is the identity element.

$\mathscr{G_{3}}$: By definition, an element is in $\mathbb{Z}_n^*$ only if it has a multiplicative inverse in $\mathbb{Z}_n^*$.

Therefore the set of elements $\bar x \in \mathbb{Z_{n}}$ such that $\bar x \bar y = \bar 1$ and $\bar y \in \mathbb{Z_{n}}$, is a group.

**(c)**

$\mathbb Z^*_8$：

· | 1 | 3 | 5 | 7 |

1 | 1 | 3 | 5 | 7 |

3 | 3 | 1 | 7 | 5 |

5 | 5 | 7 | 1 | 3 |

7 | 7 | 5 | 3 | 1 |

$\mathbb Z^*_{10}$：

· | 1 | 3 | 7 | 9 |

1 | 1 | 3 | 7 | 9 |

3 | 3 | 9 | 1 | 7 |

7 | 7 | 1 | 9 | 3 |

9 | 9 | 7 | 3 | 1 |

$\mathbb Z^*_{12}$:

· | 1 | 5 | 7 | 11 |

1 | 1 | 5 | 7 | 11 |

5 | 5 | 1 | 11 | 7 |

7 | 7 | 11 | 1 | 5 |

11 | 11 | 7 | 5 | 1 |

**(d)**

$\mathbb{Z^{*}_{8}} \simeq \mathbb{Z^{*}_{12}}$

In $\mathbb{Z^{*}_{8}}$ and $\mathbb{Z^{*}_{12}}$, each element is its own inverse, but in $\mathbb{Z^{*}_{10}}$ this is not true. Therefore $\mathbb{Z^{*}_{10}}$ is not isomorphic to $\mathbb{Z^{*}_{8}}$ and $\mathbb{Z^{*}_{12}}$.