Groups of Prime Units

Abstract

We study groups of units modulo $n$ whose nonidentity elements are all primes in $\{1,\ldots,n-1\}$. We prove that there are only nine groups of prime units which are the full group of units modulo $n$, and there are fewer than $10^{22}$ groups of prime units containing all prime units in the possible range $(\sqrt n,n-\sqrt n)\cup\{n-1\}$. We prove under Dickson's conjecture that there are infinitely many groups of prime units of arbitrary order $d \ge 2$ modulo composite integers. We prove under Schinzel's Hypothesis H that there are infinitely many groups of prime units of order 3 modulo a prime $r$. We prove that the number of such prime moduli $r$ smaller than $X$ is $\ll X/(\log X)^3$.