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

TL;DR

The paper determines a universal profile for the invariant factors of the multiplicative group $M_n=(\mathbb{Z}/n\mathbb{Z})^\times$, showing that for almost all $n$ the invariant factors follow a sequence of universal orders $\mathfrak{f}_i$, with precise asymptotic counts. It builds a bridge from group-theoretic invariant factors to additive functions, then applies Kubilius-type probabilistic methods to obtain multivariate normal (and skew-normal or related) limit laws for the counts $\mathfrak{I}(m;d)$ of factors equal to $\mathbb{Z}_d$, including explicit normalizers and special-case distributions (e.g., $d=12$). The work introduces the notion of $y$-typicality to manage typical behavior and proves density-1 results for the universal factor profile, alongside detailed average and distributional results for the invariant-factor counts. The findings deepen understanding of how arithmetic structure governs the factorization of unit groups modulo integers and have connections to Erdős–Kac-type phenomena and Dirichlet-density combinatorics. Overall, the paper provides a complete probabilistic framework describing when and how the invariant-factor decomposition of $M_n$ assumes a universal, density-1 profile and quantifies the distribution of its constituent cyclic factors.

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 $φ(n)$, the Carmichael function $λ(n)$, and the number $ω(n)$ 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 Erdős-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.

Figures (2)

• Figure 1: The universal profile of invariant factors for the multiplicative groups $(\mathbb{Z}/m\mathbb{Z})^\times$.
• Figure 2: The universal factor orders in the doubleton sets $\mathfrak{f}_i$ serve as transitions between the more numerous universal factor orders in the singleton sets.

Theorems & Definitions (161)

• Theorem 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Definition 1.5
• Theorem 1.6
• Theorem 1.7
• Example
• Theorem 1.8
• Example