FINITE AUTOMORPHISM GROUPS OF TORSION-FREE ABELIAN GROUPS OF FINITE RANK

Abstract

Abelian torsion-free groups of finite rank with finite automorphism groups are considered as rigid extensions of a system of strongly indecomposable groups Aj, j = 1, ..., k, of finite rank and having finite automorphism groups, by a finite p-group P. Such groups are called (A, p)-groups. The author introduces for (A, P)-groups the concept of (A, P)-type, which represents a choice of k integer matrices. A complete description of (A, P)-groups is given by means of (A, P)-types. Using this description, a series of problems on finite groups of automorphisms of torsion-free abelian groups of finite rank are solved. Furthermore, it is shown that the actual solution of any one of these problems comes down to a question of the consistency of a system of equations of the first degree modulo pt, where pt is the maximal order of elements of P. Bibliography: 11 titles.