Endomorphism rings and automorphism groups of almost completely decomposable groups.

*(English)*Zbl 0952.20043The authors examine the structure of the endomorphism rings and automorphism groups of completely decomposable and almost completely decomposable groups. Almost completely decomposable groups are defined as finite extensions of completely decomposable groups and both of these classes of groups are assumed to be finite rank torsion free Abelian groups throughout the discussion. Several properties are established for completely decomposable groups and are used to show that the same or nearly the same property holds for almost completely decomposable groups. For instance, (1) the nil radical of the endomorphism ring, (2) necessary and sufficient conditions for the commutativity of the endomorphism ring, and (3) necessary and sufficient conditions for the automorphism group to be torsion are all found for completely decomposable groups, and then used to establish the same property for almost completely decomposable groups. The Jacobson radical and necessary and sufficient conditions for the commutativity of the automorphism group are found for completely decomposable groups, but these results only lead to partial information in the almost completely decomposable case.

Many of these items are proven using an exact sequence for endomorphism rings which involves the type subgroups of the group. The completely decomposable groups can be characterized by an exactness property of this sequence, and they may also be characterized by the corresponding sequence for automorphism groups.

The final section of this work addresses a subclass of almost completely decomposable groups known as rigid crq-groups in order to obtain conclusive results and examples of a primarily number theoretic nature.

Many of these items are proven using an exact sequence for endomorphism rings which involves the type subgroups of the group. The completely decomposable groups can be characterized by an exactness property of this sequence, and they may also be characterized by the corresponding sequence for automorphism groups.

The final section of this work addresses a subclass of almost completely decomposable groups known as rigid crq-groups in order to obtain conclusive results and examples of a primarily number theoretic nature.

Reviewer: Ross P.Abraham (Brookings)

##### MSC:

20K15 | Torsion-free groups, finite rank |

20K30 | Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups |

20F28 | Automorphism groups of groups |

16S50 | Endomorphism rings; matrix rings |