When is the group of units in the integers modulo n cyclic?

It is easy to see with the help of Bezout’s identity that the integers co-prime to $n$ from the set $\{0,1,\cdots, n-1\}$ form a group under multiplication modulo $n$. This group is denoted by $(\mathbb{Z}/n\mathbb{Z})^*$ and it’s order is given by the Euler’s Totient function: $\varphi(n) = |(\mathbb{Z}/p\mathbb{Z})^*|$. Gauss showed that $(\mathbb{Z}/n\mathbb{Z})^*$ is a cyclic group if and only if $n=1,2,4,p,p^k$ or $2p^k$, where $p$ is an odd prime and $k > 0$. It is very easy to verify this for $n=1,2$ and $4$, as one can simply list out all positive integers less than and co-prime to $n$. So, we will only focus on proving that $(\mathbb{Z}/n\mathbb{Z})^*$ is cyclic for the remaining three cases.

Theorem 1: $(\mathbb{Z}/p\mathbb{Z})^*$ is cyclic if $p$ is a prime.

Proof: Let $d_1, d_2, \cdots , d_r$ be all the possible orders of the elements of $(\mathbb{Z}/p\mathbb{Z})^*$. Let $e=\text{lcm}(d_1, d_2, \cdots, d_r)$ and factor $e=p_1^{a_1} p_2^{a_2}\cdots p_k^{a_k}$ as a product of distinct prime powers. By the definition of $\text{lcm}$, for each $p_i^{a_i}$ there is some $d_j$ divisible by it. Since the $d_j$‘s are orders of elements of $(\mathbb{Z}/p\mathbb{Z})^*$, there is an element $x_i$ whose order is $p_i^{a_i} t$. Then, the element $y_i = x_i^t$ has order $p_i^{a_i}$. Hence, $y_1 y_2\cdots y_k$ has order $e$. Thus, we have found an element of order $e$. Therefore, $e|p-1$. But the polynomial $x^e-1$ has $p-1$ roots $(\text{mod }p)$. Since $\mathbb{Z}/p\mathbb{Z}$ is a field, any polynomial of degree $e$ cannot have more than $e$ roots. Therefore, $p-1\leq e$ and we deduce that $e=p-1$. Thus, we have found an element of order $p-1$.

Theorem 2: $(\mathbb{Z}/p^a \mathbb{Z})^*$ is cyclic for any $a\geq 1$ if $p$ is an odd prime.

Proof: For $a=1$, we are done by Theorem 1. Let $g$ be a primitive root $(\text{mod }p)$. We first find a $t$ such that $(g+pt)^{p-1}\not\equiv 1 \; (\text{mod }p^2)$. If $g^{p-1}\not\equiv 1 \; (\text{mod }p^2)$, then we can take $t=0$. Otherwise, we can choose $t=1$. Indeed, we have $(g+p)^{p-1}\equiv 1+p(p-1) g^{p-2} \not \equiv 1 \; (\text{mod }p^2)$. Let $g+pt$ have order $d\; (\text{mod }p^a)$. Then, $d|p^a (p-1)$. Since $g$ is a primitive root modulo $p$, $(p-1) | d$. So, $d=p^{r-1} (p-1)$ for some $r\leq a$. We also know that $(g+pt)^{p-1} = 1+p s$ where $s \nmid p$. Thus, \begin{aligned} (g+pt)^{p(p-1)} &= (1+ps)^{p} \\ &= \sum_{i=0}^{p} \binom{p}{i} (ps)^i \\ &\equiv 1+p^2 s \; (\text{mod }p^3) \end{aligned} By induction, it follows that $$(g+pt)^{p^{b-1}(p-1)} \equiv 1+p^b s \; (\text{mod }p^{b+1})$$ Now, $g+pt$ has order $d=p^{r-1}(p-1)\; (\text{mod } p^a)$ which implies $(g+pt)^{p^{r-1}(p-1)} \equiv 1 \; (\text{mod }p^a)$. But then $1+p^r s \equiv 1 \; (\text{mod } p^{r+1})$ if $r\leq a-1$, which implies $p | s$, a contradiction. Thus, $r=a$.

Theorem 3: $(\mathbb{Z}/2p^a \mathbb{Z})^*$ is cyclic for any $a\geq 1$ if $p$ is an odd prime.

Proof: This follows immediately from Theorem 2 and the Chinese remainder theorem.

The fact that $(\mathbb{Z}/n\mathbb{Z})^*$ can be cyclic only for the cases discussed above can also be verified easily using the Chinese remainder theorem.

References

• M. Ram Murty. Problems in analytic number theory. Springer New York, 01-Nov-2008