Abstract:
Abelian torsion-free groups of finite rank with finite automorphism groups are considered as rigid extensions of a system of strongly indecomposable groups $A_j$, $j=1,\dots,k$, 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 $p^t$, where $p^t$ is the maximal order of elements of $P$.
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