Automorphisms of Finite Abelian Groups

Much less known, however, is that there is a description of Aut(G), the automorphism group of G. The first compete characterization that we are aware of is contained in a paper by Ranum [1] near the turn of the last century. Beyond this, however, there are few other expositions. Our goal is to fill this gap, thereby providing a much needed accessible and modern treatment. Our characterization of Aut(G) is accomplished in three main steps. The first observation is that it is enough to work with the simpler groups Hp. This reduction is carried out by appealing to a fact about product automorphisms for groups with relatively prime numbers of elements (Lemma 2.1). Next, we use Theorem 3.3 to describe the endomorphism ring of Hp as a quotient of a matrix subring of Z . And finally, the units Aut(Hp) ⊂ End(Hp) are identified from this construction. As a consequence of our investigation, we readily obtain an explicit formula for the number of elements of Aut(G) for any finite Abelian group G (see also [3]).