The universal profile of the invariant factors of (ℤ/mℤ)×

The structure of the multiplicative group Formula: see text encodes a great deal of arithmetic information about the integer Formula: see text (examples include Formula: see text, the Carmichael function Formula: see text, and the number Formula: see text of distinct prime factors of Formula: see text). We examine the invariant factor structure of Formula: see text for typical integers Formula: see text, that is, the decomposition Formula: see text where Formula: see text. We show that almost all integers have asymptotically the same invariant factors for all but the largest factors; for example, asymptotically Formula: see text of the invariant factors equal Formula: see text, asymptotically Formula: see text of them equal Formula: see text, asymptotically Formula: see text of them equal Formula: see text, and so on. Furthermore, for positive integers Formula: see text, we establish a theorem of Erdős–Kac type for the number of invariant factors of Formula: see text that equal Formula: see text, except that the distribution is not necessarily a normal distribution but can also be a skew-normal or related distribution.