The Group of Units

The group of units in the integers modulo n

In the previous sections, we’ve already seen that the set Zn=Z/nZ\mathbb{Z}_{n}=\mathbb{Z} / n \mathbb{Z} consists of the elements {0,1,2,,n1}\{0,1,2, \ldots, n-1\}, whereas Zn\mathbb{Z}_{n} under addition modulo nn (see this definition :Addition_and_Multiplication_Modulo ) forms an abelian group with nn elements. We can also multiply elements of Zn\mathbb{Z}_{n}, but we don’t obtain a group necessarily for Zn\mathbb{Z}_{n} under multiplication modulo nn. For instance, the element 00 doesn’t have a multiplicative inverse. Or considering the set Z6\mathbb{Z}_{6}, whose multiplication table is illustrated in 2nd example of this lemma :Lemma_2_4_4 , we see that 1 fulfills the axiom of the identity element, but we also immediately observe that there aren’t existing inverse elements for 2,32,3, and 44.

Consequently, there’s no inverse for every element of Zn\mathbb{Z}_{n} in general. According to this definition:Addition_and_Multiplication_Modulo, the multiplication in Zn\mathbb{Z}_{n} is given by (a+nZ)(a+n \mathbb{Z}).

(b+nZ)=(ab)+nZ(b+n \mathbb{Z})=(a \cdot b)+n \mathbb{Z}. Therefore, for any n=kln=k l with integers k>1k>1 and l>1l>1, it holds that

(k+nZ)(l+nZ)=(kl)+nZ=n+nZ=0+nZ=0.(1)(k+n \mathbb{Z}) \cdot(l+n \mathbb{Z})=(k \cdot l)+n \mathbb{Z}=n+n \mathbb{Z}=0+n \mathbb{Z}=0. \quad\quad (1)

But if Zn\mathbb{Z}_{n} under multiplication would form a group in general, there would’ve to exist an inverse element k1k^{-1} for kk, such that

(k1+nZ)(k+nZ)=1+nZ.\left(k^{-1}+n \mathbb{Z}\right) \cdot(k+n \mathbb{Z})=1+n \mathbb{Z}.

Hence, we multiply both sides of equation (1) by (k1+nZ)\left(k^{-1}+n \mathbb{Z}\right), which yields.

(k1+nZ)(k+nZ)(l+nZ)=(k1+nZ)0(1+nZ)(l+nZ)=0(l+nZ)=0.\begin{aligned} \left(k^{-1}+n \mathbb{Z}\right) \cdot(k+n \mathbb{Z}) \cdot(l+n \mathbb{Z}) &=\left(k^{-1}+n \mathbb{Z}\right) \cdot 0 \\ (1+n \mathbb{Z}) \cdot(l+n \mathbb{Z}) &=0 \\ (l+n \mathbb{Z}) &=0. \end{aligned}

which contradicts to 1<l<n1<l<n. As a result, there’s no inverse for kk in general, and hence Zn\mathbb{Z}_{n} under multiplication is not necessarily a group for any nN.n \in \mathbb{N} .

However, it’s possible to enforce a group structure if we constrain our attention only to the elements of Zn\mathbb{Z}_{n} that have multiplicative inverses. As a result, we get a group under multiplication mod nmod \space n, that is often referred to as being the multiplicative group of integers modulo nn (or the group of nonzero congruence classes modulo nn ), which we call the group of units in Zn\mathbb{Z}_{n}, denoted by UnU_{n}.

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