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Regulating subgroups of Butler groups. (English) Zbl 0801.20029
Fuchs, Laszlo (ed.) et al., Abelian groups. Proceedings of the 1991 Curaçao conference. New York: Marcel Dekker, Inc.. Lect. Notes Pure Appl. Math. 146, 209-217 (1993).
Let $$A$$ be a Butler group and $$t$$ a type. Then $$A(t) = A_ t \oplus A^*(t)_ *$$ where $$A(t)$$ and $$A^*(t)_ *$$ are the usual type subgroups and $$A_ t$$ is a completely decomposable $$t$$-homogeneous group. If a group $$A_ t$$ is chosen for each type $$t$$, then $$U = \sum_ t A_ t$$ is a regulating subgroup of $$A$$. It is shown in two ways that regulating subgroups are subgroups of finite index and that there are only finitely many regulating subgroups. It follows that the intersection of all regulating subgroups of a Butler group $$A$$, the regulator $$R(A)$$ also has finite index in $$A$$. If the Butler group $$A$$ is in fact almost completely decomposable, then its regulating subgroups are completely decomposable, they are all isomorphic and have the same index in $$A$$ (Lady).
In contrast, a Butler group is constructed having non-isomorphic regulating subgroups, one of which is properly contained in the other. The regulator of an almost completely decomposable group is completely decomposable (Burkhardt) and hence its own regulator. While regulating subgroups of Butler groups are again Butler groups, little is known about their structure, in particular, it is unknown whether the chain of iterated regulators $$R(A) \supset R(R(A)) \supset \cdots$$ becomes constant after finitely many steps.
A Butler group $$A$$ is called torsionless or a $$B_ 0$$-group if $$A^*(t) = A^*(t)_ *$$ for all types $$t$$, and $$A$$ is regulating torsionless if every regulating subgroup of $$A$$ is torsionless. In a regulating torsionless Butler group $$G$$, all regulating subgroups have the same “regulating index” $$i(G)$$ as it is the case in almost completely decomposable groups. In an earlier paper Frey and Mutzbauer proved a formula for the number of regulating subgroups in an almost completely decomposable group involving the regulating index. This formula is shown to be valid for regulating torsionless Butler groups as well.
For the entire collection see [Zbl 0778.00023].