5: The Group of Units

Definition 5.1:

Let \(n \ge 2\). An element \(a \in \mathbb{Z}_n\) is said to be a unit if there is an element \(b \in \mathbb{Z}_n\) such that \(ab =1\). Here the product is multiplication modulo \(n\). We denote the set of all units in \(\mathbb{Z}_n\) by \(U_n\).

Note that 2 is a unit in \(\mathbb{Z}_5\) since \(2 \cdot 3=1\). Since the multiplication is commutative, 2 and 3 are both units. We say that 2 and 3 are inverses of each other. But note that if we write \(2^{-1}=3\), we must keep in mind that by \(2^{-1}\) in this context we do not mean the rational number \(1/2\). The following theorem is easy to prove if we assume that multiplication modulo \(n\) is associative and commutative.

Theorem \(\PageIndex{1}\)

\(U_n\) is a group under multiplication modulo \(n\). \(\blacksquare\)

We call \(U_n\) the group of units of \(\mathbb{Z}_n\).

Problem 5.1 List all the elements of \(U_n\) for \(n \in \{ 2,3,4,\dots, 12\}\).

Problem 5.2 For which \(n \in \{ 2,3,4,\dots, 12\}\) is there an element \(a \in U_n\) such that \(U_n = \langle a \rangle\)?

Theorem \(\PageIndex{2}\)

For \(n\geq 2\), \(U_n = \{ a\in \mathbb{Z}_n: \text{gcd}(a,n)=1\}. \blacksquare\)

Remark

This theorem is established in number theory courses. In number theory, the order of the group \(U_n\) is important enough to have its own name and notation. The order of \(U_n\) is denoted by \(\phi(n)\), is called the Euler totient function and is pronounced fee of n. In number theory it is proved that if \(a\) and \(b\) are positive integers such that \(\gcd(a,b)=1\) then \(\phi(ab) = \phi(a)\phi(b)\) and if \(p\) is prime and \(n \in \mathbb{N}\) then \(\phi(p^n) = p^n - p^{n-1}\). These facts make it easy to compute \(\phi(n)\) if one can write \(n\) as a product of primes. But there is no known easy way to compute \(\phi(n)\) if the factorization of \(n\) is unknown.

Note that there are four different but similar symbols used in mathematics:

1. \(\phi\) : lower case Greek letter phi (pronounced fee)
2. \(\Phi\) : capital Greek letter Phi
3. \(\varphi\) : lower case script Greek letter phi
4. \(\emptyset\) : slashed zero (not Greek, but Danish) and symbol for the empty set

Problem 5.3 Prove the easy part of Theorem 5.2; namely, show that if \(a \in \mathbb{Z}_n\) and \(\gcd(a,n)=d > 1\), then \(a\) is not a unit. [Hint: Show (1) that if \(a \in \mathbb{Z}_n\) and \(\gcd(a,n)=d > 1\) there is an element \(b \in \mathbb{Z}_n-\{ 0 \}\) such that \(ab=0\). (2) If \(b \in \mathbb{Z}_n -\{ 0\}\) and \(ab=0\) then \(a\) is not a unit. ]

Theorem \(\PageIndex{3}\)

If \(p\) is a prime then there is an element \(a \in U_p\) such that \(U_p = \langle a \rangle\). \(\blacksquare\)

Proof. This theorem is proved in advanced courses in number theory or abstract algebra.

Demonstrate Theorem 5.3 for all primes \(p < 12\).

Remark

It will be noted that sometimes even when \(n\) is not prime there is an \(a \in U_n\) such that \(U_n = \langle a \rangle\). In fact, the following theorem from advanced number theory tells us exactly when such an \(a\) exists.

Theorem \(\PageIndex{4}\)

If \(n \ge 2\) then \(U_n\) contains an element \(a\) satisfying \(U_n = \langle a \rangle\) if and only if \(a\) has one of the following forms: 2, 4, \(p^k\), or \(2p^k\) where \(p\) is an odd prime and \(k \in \mathbb{N}\). \(\blacksquare\)

So, for example, there is no such \(a\) in \(U_n\) if \(n = 2^k\) when \(k \ge 3\), nor for \(n = 12\) or \(15\).