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

The structure of the multiplicative group $M_n=(\mathbb{Z}\mathrm{/}n\mathbb{Z}{)}^{\times }$ 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 $M_n\cong \mathbb{Z}\mathrm{/}{d}_1\mathbb{Z}\times \mathbb{Z}\mathrm{/}{d}_2\mathbb{Z}\times \cdots \times \mathbb{Z}\mathrm{/}{d}_k\mathbb{Z}$ 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 $1\mathrm{/}2$ of the invariant factors equal $\mathbb{Z}\mathrm{/}2\mathbb{Z}$, asymptotically $1\mathrm{/}4$ of them equal $\mathbb{Z}\mathrm{/}12\mathbb{Z}$, asymptotically $1\mathrm{/}12$ of them equal $\mathbb{Z}\mathrm{/}120\mathbb{Z}$, 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 $\mathbb{Z}\mathrm{/}k\mathbb{Z}$, except that the distribution is not a normal distribution but rather a skew-normal or related distribution.