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