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.