Anatomy of the mulitplicative group

Greg Martin

The multiplicative group of units in the ring Z/nZ is one of the first groups an undergraduate student of algebra encounters. As n varies over the positive integers, these multiplicative groups form a naturally occurring family of finite abelian groups, whose structure encodes various interesting arithmetic invariants of the modulus n (for example, the number of prime factors of n). Their algebraic structure is already somewhat complicated, but the real fun begins when we want to understand the statistical distribution of these invariants: what distribution do we get, for example, if we choose positive integers "at random" and record how many prime factors they have? Once this question has been made rigorous, the surprising answer is a particular Gaussian distribution.

In this talk we take a tour of several interesting arithmetic invariants connected to these multiplicative groups: the largest order of any element, the number of factors in the primary decomposition and the largest such factor, the number of elements of a fixed order, and the total number of subgroups. Many of the speaker's current research projects involve the investigation of the distribution of invariants of this sort.