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# Local-global Relationships

DOI link for Local-global Relationships

Local-global Relationships book

# Local-global Relationships

DOI link for Local-global Relationships

Local-global Relationships book

## ABSTRACT

Let X be an almost completely decomposable group and A a completely decom posable subgroup of finite index. The finite group X / A has a direct decomposition into primary components and it is natural to consider the subgroups X Xp where

Obviously X contains many other completely decomposable subgroups of finite in dex besides A, for example the groups n A , and there is nothing canonical about the constructs X xp. However, there are special completely decomposable subgroups of finite index, namely the regulating subgroups and the regulator R(X) of X. Recall that the regulator is a fully invariant completely decomposable subgroup of finite index of X and both R(X) and X / R ( X ) are isomorphism invariants of the almost completely decomposable group X. Thus every almost completely de composable group contains a canonical completely decomposable subgroup of finite index, namely R(X), and the p -co n stituen ts X Xp relative to this subgroup are also canonical subgroups. In this chapter we study the relationships between an almost completely decomposable group X and its primary constituents X Xp in a systematic and general manner. Recall (Definition 4.5.8) that the primes dividing the index [X : R(X)] are called relevant primes of X . There is ample evidence (Chapter 3, Corollary 10.4.6, Chapter 12) that the theory of almost completely decomposable groups is much simpler when there is a single relevant prime; this is what we mean by the local case. One of the basic results of this chapter is that R ( X Xp) = R(X) (Theorem 5.1.3) so that considering primary constituents means passing to the lo cal case. Having answers in the local case the question becomes what they mean for the global (general) case. We consider the established and important concepts of the theory of almost completely decomposable groups and clarify their localglobal relationships. This program is remarkably successful, and the connection between the global group and its local constituents is quite close. See, for example Proposition 5.1.5, Theorem 5.2.3, Corollary 5.2.4, Proposition 5.3.1, Theorem 5.3.2, Theorem 8.1.15. However, the general connection between direct decompositions of the primary constituents and the global group remains mysterious.