(related to Proposition: Multiplicative Group Modulo an Integer $(\mathbb Z_m^*,\cdot)$)

- Let $m > 1$ be an integer and let the set $\mathbb Z_m^*$ contain only those congruence classes modulo $m$, which are co-prime to $m$.
- Creation of reduced residue systems from others demonstrates that $(\mathbb Z_m^*,\cdot)$ is closed under the multiplication operation $"\cdot".$
- Moreover, "$\cdot$" is commutative and associative (since it was in the ring $\mathbb Z_m.$
- Moreover, we have $1(m)\in\mathbb Z_m^*.$
- It follows from the existence and number of solutions of congruence with one variable that the congruence $(ax)(m)\equiv 1(m)$ has a unique solution $x:=a^{-1}(m)\in\mathbb Z_m^*$ for every $a(m)\in\mathbb Z_m^*.$
- Thus, each $a(m)\in\mathbb Z_m^*$ has a multiplicative inverse.
- Altogether, we have shown that the algebraic structure $(\mathbb Z_m^*,\cdot)$ is a commutative group.∎

**Kraetzel, E.**: "Studienbücherei Zahlentheorie", VEB Deutscher Verlag der Wissenschaften, 1981