The universal profile of the invariant factors of $({\mathbb Z}/n{\mathbb Z})^\times$

Abstract

The structure of the multiplicative group $M_n = ({\mathbb Z}/n{\mathbb Z})^\times$ encodes a great deal of arithmetic information about the integer $n$ (examples include $\phi(n)$, the Carmichael function $\lambda(n)$, and the number $\omega(n)$ of distinct prime factors of $n$). We examine the invariant factor structure of $M_n$ for typical integers $n$, that is, the decomposition $M_n \cong {\mathbb Z}/d_1{\mathbb Z} \times {\mathbb Z}/d_2{\mathbb Z} \times \cdots \times {\mathbb Z}/d_k{\mathbb Z}$ where $d_1\mid d_2\mid\cdots\mid d_k$. We show that almost all integers have asymptotically the same invariant factors for all but the largest factors; for example, asymptotically $1/2$ of the invariant factors equal ${\mathbb Z}/2{\mathbb Z}$, asymptotically $1/4$ of them equal ${\mathbb Z}/12{\mathbb Z}$, asymptotically $1/12$ of them equal ${\mathbb Z}/120{\mathbb Z}$, and so on. Furthermore, for positive integers $k$, we establish a theorem of Erd\H{o}s-Kac type for the number of invariant factors of $M_n$ that equal ${\mathbb Z}/k{\mathbb Z}$, except that the distribution is not a normal distribution but rather a skew-normal or related distribution.