Automorphisms of the Unit Groups of Square Radical Zero Finite Commutative Completely Primary Rings

Abstract

Let G be a group. The groups G ′ for which G is an automorphism group have not been fully characterized. Suppose R is a Completely Primary finite Ring with Jacobson Radical J such that J 2 = (0). In this case, the characteristic of R is p or p 2 and the group of units R ∗ = Zpr−1 × (I + J) . The structure of R ∗ is well known, but its automorphism group is not well documented. Given the group R ∗ , let Aut(R ∗ ) denote the group of isomorphisms φ : R ∗ → R ∗ with multiplication given by the composition of functions. The structure of the automorphism groups of finite groups is intimately connected to the structure of the finite groups themselves. In this note, we determine the structure of Aut(R ∗ ) using well known procedures and to this end, extend the results previously obtained in this area of research